Constitutive Equations for Superelasticity in
Crystalline Shape-Memory Materials
by
byOCT
MASSACHUSETTS rSITUTE
__OF TECHNOLOGY
2 520
Prakash Thamburaj a
B.Eng. Mechanical Engineering
Imperial College of Science, Technology and Medicine, 1996
LIBRARIES
BARKER
S.M. Mechanical Engineering
Massachusetts Institute of Technology, 2000
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2002
© Prakash Thamburaja, MMII. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author ..........................................
Department of Mechanical Engineering
August 9, 2002
Certified by........
.............
Lallit Anand
Professor of Mechanical Engineering
Thesis Supervisor
a
................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
Accepted by...
Constitutive Equations for Superelasticity in Crystalline
Shape-Memory Materials
by
Prakash Thamburaja
Submitted to the Department of Mechanical Engineering
on August 9, 2002, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
A crystal-mechanics-based constitutive model for polycrystalline shape-memory materials has been developed. The model has been implemented in a finite-element
program. Finite-element calculations of polycrystal response were performed using
two methods: (1) The full-finite element method where each element represents a
single crystal chosen from a set of crystal orientations which approximate the initial
crystallographic texture; (2) A simplified model using the Taylor assumption (1938)
where each element represents a collection of single crystals at a material point. The
macroscopic stress-strain responses are calculated as volume averages over the entire
aggregate.
A variety of superelastic experiments were performed on initially-textured Ti-Ni
rods and sheets. The predicted stress-strain curves from finite-element calculations
are shown to be in good accord with the corresponding experiments.
For the Ti-Ni sheet, strain-temperature response at a fixed stress was also experimentally studied. The model was also shown to accurately predict the results
from these important experiments. Further, by performing superelastic experiments
at moderately high strain rates, the effects of self-heating and cooling due to the
phase transformations are shown to be captured well by the constitutive model. The
thermo-mechanically-coupled theory is also able to capture the resulting inhomogeneous deformations associated with the nucleation and propagation of transformation
fronts.
Finally, an isotropic constitutive model has also been developed and implemented
in a finite-element program. This simple model provides a reasonably accurate and
computationally-inexpensive tool for purposes of engineering design.
Thesis Supervisor: Lallit Anand
Title: Professor of Mechanical Engineering
2
Acknowledgments
First and foremost, I would like to thank my advisor Prof. Lallit Anand for his
support. His high academic standards and great vision will always be a source of
inspiration to me. I would also like to thank Prof. Mary Boyce, Prof. David Parks
and Prof. William Carter for serving on my committee.
We are also thankful to Gao Shan and Prof. Yi Sung of the Nanyang Technological
University, Singapore for supplying us with the raw material and the experimental
results for their Ti-Ni sheet.
To the Mechanics and Materials group - Nicoli Ames, Jeremy Levitan, Matt
Busche, Jin Yi, Cheng Su, Theodora Tzianetopoulou, Hang Qi, Rajdeep Sharma,
Mats Danielsson, Ethan Parsons, Franco Capaldi and others, thank you very much
for all the great conversations and good times we have had. I wish all of you the best
of luck in your future endeavors.
To Ray Hardin and Leslie Regan, thank you very much for taking care of all the
administrative details and providing me with your unerring advice.
To Brian P. Gearing, my colleague and most of all friend for the last six years,
thank you very much for all that you have done for me. May success shine on you
constantly.
To my family, Drs. Thamburaja and Surabi, Sree Priya and Jeyalakshmi, your
constant love, affection and support have been very instrumental throughout my life
journey thus far. Thank you very much for everything.
The financial support for this work was provided by the National Science Foundation under Grant CMS0002930.
3
Contents
1
Introduction
16
2
Crystal-mechanics-based constitutive model
29
2.1
Single-crystal constitutive equations . . . . . . . . . . . . . . . . . . .
2.2
Evaluation of the constitutive model for a polycrystalline Ti-Ni alloy :
2.3
29
R od-texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.2.1
Taylor model
45
2.2.2
Effect of temperature on the deformation of polycrystalline Ti-Ni 46
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Evaluation of the constitutive model for a polycrystalline Ti-Ni alloy :
Sheet-texture
48
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3 Isotropic-based constitutive model
3.1
Constitutive m odel . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.2
Evaluation of the constitutive model for polycrystalline Ti-Ni alloy . .
88
3.2.1
Sheet m aterial . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.2.2
Rod m aterial
94
3.3
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Deformation of a Ti-Ni biomedical stent
4 Conclusion and future work
. . . . . . . . . . . . . . . .
96
115
A Time-integration procedure : Crystal-mechanics-based constitutive
model
123
B Crystallographic theory of martensite
4
128
C Algorithm to calculate the austenite-martensite transformation systems using the crystallographic theory of martensite
135
C.1 Results of the CTM theory . . . . . . . . . . . . . . . . . . . . . . . . 137
D Example of a fortran code to calculate the austenite-martensite
transformation systems using CTM
146
E Time-integration procedure : Isotropic-based constitutive model
185
F Hypo-elastic form for the isotropic-based constitutive equations
189
F.1
Increm ental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
F.2
Time integration procedure
F.3
Jacobian M atrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
. . . . . . . . . . . . . . . . . . . . . . .
197
G Simple shear problem in the isotropic-based constitutive model
203
H Anisotropic superelasticity of textured sheet Ti-Ni
206
I
Operating procedure for the RigaKu200 & 300 X-ray Diffraction
machine
217
5
List of Figures
1-1
(a) Shape-memory alloy-based actuation device (Baron et al., 1994).
(b) Scanning Electron Microscope (SEM) image of a Scimedt"' artery
stent (Serruys, 1997).
1-2
. . . . . . . . . . . . . . . . . . . . . . . . . .
24
Schematic diagram of the (a) superelastic, and (b) strain-temperature
cycling response. A-M denotes the austenite-to-martensite transformation whereas M-A denotes the martensite-to-austenite transformation.
1-3
25
Experimental stress-strain curves for Ti-Ni rod at 0 = 298 K in simple
tension and simple compression. For compression the absolute values
of stress and strain are plotted. . . . . . . . . . . . . . . . . . . . . .
26
1-4 Experimental stress-strain curves for Ti-Ni sheet at 0 = 300 K in simple
tension along the rolling and transverse direction
1-5
. . . . . . . . . . .
27
Experimental stress-strain curves for a Ti-Ni sheet at 0 = 300 K in
simple tension at strain rates of 0.00003/sec, 0.00084/sec, and 0.002/sec. 28
2-1
(a) Experimentally-measured texture in the as-received Ti-Ni rod, (b)
its numerical representation using 729 unweighted discrete crystal orientations, and (c) its numerical representation using 343 unweighted
discrete crystal orientations. Pole-figure data were obtained using PoPLa. 55
2-2
Differential scanning calorimetry thermogram for polycrystalline rod
Ti-Ni used in experiments. . . . . . . . . . . . . . . . . . . . . . . . .
2-3
Schematic of an isothermal superelastic stress-strain response in simple
tension .
2-4
56
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Specimen geometry for tension and compression experiments . . . . .
58
6
2-5
(a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed
mesh at a tensile strain of 6%. Contours of martensite volume fraction are shown. (c) Superelastic stress-strain curve in tension. The
experimental data from this test was used to estimate the constitutive
parameters. The curve fit using the full finite element model of the
polycrystal is also shown. . . . . . . . . . . . . . . . . . . . . . . . . .
2-6
59
(a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed
mesh at a compressive strain of 5%. Contours of martensite volume
fraction are shown. (c) Superelastic stress-strain curve in compression.
The absolute values of stress and strain are plotted. The prediction
from the full finite-element model for the polycrystal is also shown.
2-7
.
60
Comparison of the predicted response from tension and compression simulations to demonstrate the numerically predicted tensioncompression asymmetry for (a) the polycrystal material and (b) single
crystal oriented in the {111} direction. . . . . . . . . . . . . . . . . .
2-8
61
(a) {111} pole figure corresponding to a random initial texture. (b)
Comparison of the predicted stress-strain response in tension and com-
2-9
pression using a random initial texture. . . . . . . . . . . . . . . . . .
62
Specimen geometry for torsion experiments.
63
. . . . . . . . . . . . . .
2-10 (a) Undeformed mesh of 720 ABAQUS C3D8R elements. (b) Deformed
mesh at a shear strain of 9%. Contours of martensite volume fraction
are shown.
(c) Superelastic stress-strain curve in torsion. The pre-
diction from the full finite-element model for the polycrystal is also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
64
2-11 (a) Loading program for combined tension-torsion experiment.
(b)
Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed
mesh at a tensile strain of 3% and a shear strain of 4.5%. Contours
of martensite volume fraction are shown. (d) Superelastic stress-strain
curve in tension. (e) Superelastic stress-strain curve in shear. The
prediction from the full finite element model for the polycrystal is also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
2-12 (a) Loading program for the path change tension-torsion experiment.
(b) Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed
mesh at a tensile strain of 3% and a shear strain of 7%. Contours of
martensite volume fraction are shown. (d) Superelastic stress-strain
curve in tension. (e) Superelastic stress-strain curve in shear. The
prediction from the full finite element model for the polycrystal is also
shown.........
....................................
66
2-13 Comparison of the Taylor model against experiment and full finiteelement calculation in (a) simple tension, (b) simple compression, and
(c) shear.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2-14 Response of the Taylor Model compared against the path change
tension-torsion experiment and full finite element calculation in (a)
tension and (b) shear.
. . . . . . . . . . . . . . . . . . . . . . . . . .
68
2-15 Superelastic stress-strain curves in tension (Shaw and Kyriakides,
1995) at three temperatures (a) 9 = 333.2 K, (b) 0 = 343.2 K, and
(c) 0 = 353.2 K. Full finite-element and Taylor model prediction from
the constitutive model are also shown. The material parameters for
the Ti-Ni of Shaw and Kyriakides is obtained from the data at 343.2 K. 69
2-16 Simulation of austenite-martensite-austenite shape memory effect (Shaw
and Kyriakides, 1995): Isothermal stress-strain response for straining
at constant strain rate (ABC) at 6 = 303.2 K followed by a temperature
(CD) to 0 = 305.8 K.
. . . . . . . . . . . . . . . . . . . . . . . . . .
8
70
2-17 (a) Experimentally-measured texture in the as-received Ti-Ni sheet;
and (b) its numerical representation using 48 weighted crystal orientation s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-18 (a) Geometry of the tensile specimen (drawing is not to scale).
71
(b)
Undeformed finite-element mesh of the tensile specimen using 446
ABAQUS three-dimensional elements. The gage width of the specimen is linearly tapered from 3.85 mm to 3.65 mm as shown in (a). . .
72
2-19 Schematic diagram of the variation of resolved forces on an active transformation system i with temperature. . . . . . . . . . . . . . . . . . .
73
2-20 Nominal superelastic stress-strain curves from tension tests on 0'oriented specimens. The tests were conducted under isothermal conditions at (a) 289 K, (b) 300 K, and (c) 306 K. The experimental data
from these tests were used to estimate the phase transformation parameters. The curve fit from the finite-element simulations is also shown.
74
2-21 A superelastic finite-element simulation of an experiment conducted
under isothermal conditions at 300 K. Contour plots of the martensite volume fraction are also shown. The set of contours on the left
shows the transformation from austenite-to-martensite during forward
transformation, while those on the right show the transformation from
martensite-to-austenite during reverse transformation. Note that because of the slight taper in the gage width, the austenite-to-martensite
transformation nucleates at the right grip-end and propagates toward
the opposing grip-end. . . . . . . . . . . . . . . . . . . . . . . . . . .
75
2-22 Experimental superelastic stress-strain curves in tension at 300 K: (a)
450 orientation; (b) 90' orientation.
The corresponding predictions
from the finite-element simulations are also shown.
. . . . . . . . . .
76
2-23 Strain-temperature cycling experiments conducted on 0-oriented specimens at pre-stress levels of (a) 150 MPa; and (b) 250 MPa. The predictions from the finite-element simulations are also shown.
9
. . . . .
77
2-24 Superelastic stress-strain curves in tension along the rolling direction at
nominal strain rates of 2 x 10- /sec, 8.4 x 10-4 /sec, and 3 x 10- /sec
("isothermal condition").
The predictions from the theory are also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
2-25 A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10- /sec. The set of the martensite volume
fraction contours on the left show the transformation from austeniteto-martensite during forward transformation, while those on the right
show the transformation from martensite-to-austenite during reverse
transformation. Note that because of the boundary conditions, both
the forward and reverse transformations nucleate from the grip-ends
and propagate towards the center of the specimen. . . . . . . . . . . .
79
2-26 A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10- /sec. The set of contours on the left
shows the temperature increase during transformation from austeniteto-martensite, while those on the right show the temperature decrease
during transformation from martensite-to-austenite.
During forward
transformation the temperature increases by as much as 22'K above
the ambient temperature of 300'K due to the exothermic austenite-tomartensite transformation, whereas it decreases by as much as 22'K
below the ambient temperature during the reverse endothermic transformation from martensite-to-austenite.
. . . . . . . . . . . . . . . .
80
2-27 (a) Nominal strain versus time profile for the tension-hold experiment
at a baseline nominal strain rate of 1.25 x 10- /sec and initial temperature of 300 K. (b) Superelastic stress-strain curve measured in the
experiment. The prediction from the finite-element simulation is also
shown.
........
..................................
10
81
3-1
Comparison of the implicit and explicit finite-element solution of the
tension specimen shown in Figure 2-18b at a tensile strain-rate of
0.002/s at an initial temperature of 0 = 300 K using the material param eters in Section 3.2.1. . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
99
(a) Schematic of an isothermal superelastic stress-strain response in
simple tension. (b) Schematic diagram of the variation of the nucleation stresses -am and -ma with temperature.
3-3
. . . . . . . . . . . . .
100
Nominal superelastic stress-strain curves from tension tests on 0'oriented specimens. The tests were conducted under isothermal conditions at (a) 289 K, (b) 300 K, and (c) 306 K. The experimental data
from these tests were used to estimate the phase transformation parameters. The curve fit from the finite-element simulations is also shown. 101
3-4
A superelastic finite-element simulation of an experiment conducted
under isothermal conditions at 300 K. Contour plots of the martensite volume fraction are also shown. The set of contours on the left
shows the transformation from austenite-to-martensite during forward
transformation, while those on the right show the transformation from
martensite-to-austenite during reverse transformation. Note that because of the slight taper in the gage width, the austenite-to-martensite
transformation nucleates at the right grip-end and propagates toward
the opposing grip-end. . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3-5
Strain-temperature cycling experiments conducted on 0W-oriented specimens at pre-stress levels of (a) 150 MPa; and (b) 250 MPa. The predictions from the finite-element simulations are also shown.
3-6
. . . . . 103
Superelastic stress-strain curves in tension along the rolling direction at
nominal strain rates of 2 x 10-3 /sec, 8.4 x 10-4 /sec, and 3 x 10-5 /sec
("isothermal condition").
show n .
The predictions from the theory are also
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11
3-7
A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10- /sec. The set of the martensite volume
fraction contours on the left show the transformation from austeniteto-martensite during forward transformation, while those on the right
show the transformation from martensite-to-austenite during reverse
transformation. Note that because of the boundary conditions, both
the forward and reverse transformations nucleate from the grip-ends
and propagate towards the center of the specimen. . . . . . . . . . . .
3-8
105
A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10-3 /sec. The set of contours on the left
shows the temperature increase during transformation from austeniteto-martensite, while those on the right show the temperature decrease
during transformation from martensite-to-austenite.
During forward
transformation the temperature increases by as much as 25'K above
the ambient temperature of 300 K due to the exothermic austenite-tomartensite transformation, whereas it decreases by as much as 25'K
below the ambient temperature during the reverse endothermic transformation from martensite-to-austenite. . . . . . . . . . . . . . . . . .
3-9
106
(a) Nominal strain versus time profile for the tension-hold experiment
at a baseline nominal strain rate of 1.25 x 10- /sec and initial temperature of 300 K. (b) Superelastic stress-strain curve measured in the
experiment. The prediction from the finite-element simulation is also
shown. ........
..................................
107
3-10 (a) Superelastic stress-strain curve in tension. The experimental data
from this test were used to estimate the constitutive parameters. The
curve fit using the constitutive model is also shown, and (b) Superelastic stress-strain curve in torsion. The prediction from the constitutive
m odel is also shown.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
12
3-11 (a) Loading program for combined tension-torsion experiment.
(b)
Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed
mesh at a tensile strain of 3% and a shear strain of 4.5%. Contours
of martensite volume fraction are shown. (d) Superelastic stress-strain
curve in tension. (e) Superelastic stress-strain curve in shear. The
prediction from the constitutive model is also shown.
. . . . . . . . .
109
3-12 (a) Illustration of the SciMed stent. The 14-mm stent consists of five
segments attached to each other at three sites. This design allows
for good support without gaps, and high flexibility, and (b) Scanning
electronic microscopic picture of the SciMed stent with full expansion.
(Taken from the Handbook of coronary stents, (Serruys, 1997)).
. . .
110
3-13 (a) Geometry of diamond-shaped strut test specimens; all dimensions
are in millimeters. The sheet form which the specimens were machined
is 0.53 mm thick. (b) Initial finite-element mesh of one-quarter of the
specimen.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
3-14 Superelastic force-displacement curve of the diamond-shaped strut test
specimen in tension and compression. The corresponding prediction
from the finite-element simulation is also shown. . . . . . . . . . . . . 112
3-15 (a) The physical position of the diamond-section stent, and (b) the corresponding finite-element mesh with contours of the martensite volume
fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
3-16 (a) Initial finite-element mesh of a segment of the stent.(b) The fully
crimped state of the stent with contours of martensite volume fraction.
(c) Predicted axial load-displacement curve from the finite-element
sim ulation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B-i (a) Schematic diagram of an austenite-twinned martensite transformation system, and (b) a magnified schematic diagram of the twinned
m artensite.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
134
G-1 Comparison of the shear-stress versus shear-strain response in simple
shear predicted by the non-standard flow rule versus that predicted by
the standard-Mises type flow rule. The nonstandard flow rule predicts
the expected closed flag-type hysteresis loop.
. . . . . . . . . . . . . 205
H-1 (a) Experimentally-measured texture in the as-received Ti-Ni sheet,
and (b) its numerical representation using 420 discrete unweighted
crystal orientations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 209
H-2 (a) Undeformed mesh of 420 ABAQUS C3D8R elements used to represent a textured polycrystal aggregate. (b) Superelastic stress-strain
curve in tension along the rolling direction. The experimental data
from this test were used to estimate the constitutive parameters. The
curve fit using the full finite-element model of the polycrystal is also
shown........
....................................
210
H-3 (a) Superelastic experiment conducted along the (a) 100 and (b) 20'
direction. The prediction from the finite-element simulations are also
show n .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
H-4 (a) Superelastic experiment conducted along the (a) 30' and (b) 40'
direction. The prediction from the finite-element simulations are also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
H-5 (a) Superelastic experiment conducted along the (a) 500 and (b) 60'
direction. The prediction from the finite-element simulations are also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
213
H-6 (a) Superelastic experiment conducted along the (a) 70' and (b) 80'
direction. The prediction from the finite-element simulations are also
show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
14
H-7 (a) Superelastic experiment conducted along the transverse direction.
The prediction from the finite-element simulations is also shown, and
(b) Superelastic experiment along the rolling direction compared with
the transverse direction. The finite-element simulation along the transverse direction is also shown. . . . . . . . . . . . . . . . . . . . . . . . 215
H-8 Comparison of the nucleation stresses of the differently-oriented specimens during the forward and reverse transformation at 2% strain with
respect to the prediction from the finite-element simulations. . . . . . 216
15
Chapter 1
Introduction
Shape-memory alloys (SMAs) are finding increased use as functional/smart-materials
for a variety of applications (see Figure 1-1). The individual grains in these polycrystalline materials can abruptly change their lattice structure in the presence of suitable
thermo-mechanical loading. This capability of undergoing a solid-solid, diffusionless,
displacive phase transformation leads to the technologically-important properties of
superelasticity and shape-memory.
Currently, the near-equiatomic Ti-Ni alloys are the most popular shape-memory
materials for applications. The reversible transformations between the various phases
observed upon changing the temperature, under zero stress, in the case of Ti-Ni are
as follows. The transformation sequence upon cooling from a high-temperature bodycentered-cubic superlattice, B2, austenite phase is first to a rhombohedral R-phase,
and then to a monoclinic martensite phase. Upon heating, the reverse transformation takes place; however, the R-phase is not observed. Under certain conditions,
the R-phase may be suppressed, and the only transformation in a specimen is the
cubic-to-monoclinic transition. Under zero stress, shape-memory materials are distinguished by the following four temperatures: Oms, martensite start temperature;
9
mf, martensite finish temperature;
0
as,
austenite start temperature; and Oaf, austen-
ite finish temperature.
1Also called pseudoelasticity by transformation.
16
As mentioned previously, an important characteristic response of shape-memory
materials is superelasticity. A schematic diagram for the superelastic response is
shown in Figure 1-2a. This is a consequence of a stress-induced transformation from
austenite to martensite and back when a sample is tested in cyclic uniaxial extension,
between zero and a finite but small (~5%) strain, under quasi-static conditions at a
constant ambient temperature above its austenite finish temperature, Oaf. There is
little or no permanent deformation experienced by the specimen in such a strain cycle; this gives an impression that the material has only undergone elastic deformation,
and hence the term superelastic. However, there is hysteresis; the mechanism responsible for the hysteresis is the motion of sharp interfaces between the two material
phases. For a given material, the size and other qualitative features of the "flagtype" hysteresis loops usually change with the loading rate and the temperature at
which the test takes place. Superelastic response can also be triggered by a change
in temperature after a pre-stress is applied on the shape-memory material. This is
widely known as the strain-temperature cycling response. A schematic diagram for
the strain-temperature cycling response is shown in Figure 1-2b.
The shape-memory effect by transformation occurs when a material which is initially austenitic is first tested in isothermal uniaxial extension at a temperature
0
ms < 0 < Oaf. During forward loading, a transformation from austenite to marten-
site occurs, but upon reversal and unloading to zero stress, the transformation strain
does not recover until the temperature is subsequently raised to 0 > Oaf.
If the temperature is lower than the martensite finish temperature, 0 < Omf,
the material is initially in the martensitic state. In this condition SMAs can also
exhibit superelastic and shape-memory effects, but the inelastic strain is caused by
reorientation of the martensitic variants, and not by transformation (e.g. Saburi and
Nenno, 1981; Miyazaki and Otsuka, 1989; Otsuka and Wayman, 1999).
There is substantial activity worldwide to construct suitable constitutive models
for shape-memory materials, and several existing one-dimensional constitutive models
can capture the major response characteristics of SMAs reasonably well (e.g. Liang
and Rogers, 1990; Abeyaratne and Knowles, 1993; Ivshin and Pence, 1994; Bekker
17
and Brinson, 1997).
Prominent amongst the three-dimensional models are the ones proposed by Sun
and Hwang (1993a,b), Boyd and Lagoudas (1994,1996), Patoor et al. (1996), Auricchio and Taylor (1997), Lu and Weng (1998), Gao et al. (2000) and others, but
most of these models have been shown to work best for uniaxial loading. It is difficult to test the applicability of these models in real three-dimensional situations
because there is a lack of pedigreed multi-axial experimental data, although some
nice experiments have been recently reported by Lim and McDowell (1999). Indeed,
the predictions from these models which have been calibrated from data for simple
tension, have not even been verified for the case of simple shear. Also, these models
do not adequately capture the asymmetry observed between tension and compression
superelastic experiments, where it is found that at a given test temperature: (i) the
stress level required to nucleate the martensitic phase from the parent austenitic phase
is considerably higher in compression than in tension; (ii) the transformation strain
measured in compression is smaller than that in tension; and (iii)the hysteresis loop
generated in compression is wider (measured along the stress axis) than the hysteresis
loop generated in tension. These major differences between the tension and compression response of a Ti-Ni rod alloy in superelasticity experiments (to be described
more fully later) are shown in Figure 1-3. Anisotropic effects are even manifested in a
Ti-Ni sheet alloy, as shown in Figure 1-4 where the superelastic tensile response along
the rolling and transverse direction is shown. Further investigation on superelastic
behavior at various deformation rates also show that : (i) The stress-strain response
"hardens" more as the rate of deformation increases; and (ii) the hysteresis loops
get wider (measured along the stress axis) as the deformation rate increases. These
experimental results on a Ti-Ni sheet conducted at various deformation rates at the
same ambient temperature are shown in Figure 1-5.
It is now well recognized that shape-memory materials derive their unusual and
inherently nonlinear and anisotropic properties from the fine-scale rearrangements
of phases, or "microstructures," and that the strain produced by the superelasticity effect depends on crystal orientations. Specially-oriented single crystals of some
18
shape-memory materials can produce sizeable strains (~
10%) due to phase trans-
formations. In applications, shape-memory materials are typically polycrystalline in
nature, and are usually processed by casting, followed by hot-working (drawing for
rods and wires, and rolling for sheet) and suitable heat treatments. Polycrystalline
SMAs so produced are usually strongly textured, and various researchers have recently
emphasized that crystallographic texture is very important in determining the overall
properties of SMAs (e.g. Inoue et al., 1996; Zhao et al., 1998; Shu and Bhattacharya,
1998; Gall and Sehitoglu, 1999).
Shu and Bhattacharya (1998) have developed an analytical geometric model to
estimate the effects of initial crystallographic texture on the amount of recoverable
strains in SMAs. They show that texture is important in determining the amount of
shape-memory effect in polycrystals. In particular, they qualitatively show that even
though Ti-Ni and Cu-Zn-Al based SMAs both undergo cubic-monoclinic transformations, and both have similar transformation strains at the single-crystal level, it is
the difference in the crystallographic texture between the two polycrystalline SMAs
in bulk sheet, rod, and wire forms which gives rise to a larger shape-memory effect
in Ti-Ni.
More recently, Gall and Sehitoglu (1999) have studied the stress-strain behavior of polycrystalline Ti-Ni under tension versus compression. They used a micromechanical model which incorporates single-crystal constitutive equations and experimentally measured polycrystalline texture into a "self-consistent" model for polycrystals (Patoor et al., 1996) to argue that the tension/compression asymmetry in
Ti-Ni shape-memory alloys was related to the initial crystallographic texture of their
specimens2
An important feature of the superelastic response of Ti-Ni wires observed by Shaw
2
We do not completely understand the model and computational procedure used by Gall and
Sehitoglu (1999) because sufficient details are not provided by the authors. It appears that these
authors consider a model in which the transformation rates are determinable only if the model
contains an invertible "interaction energy matrix" (their equation 11). They attribute the magnitude
of the tension-compression asymmetry in the superelasticity behavior of their Ti-Ni material to the
magnitudes of the terms of the interaction moduli. In contrast, we show in this work that the major
features of the tension-compression asymmetry in textured drawn bars of Ti-Ni are captured even
if the interaction matrix is set to zero.
19
and Kyriakides (1997) was that the transformation from austenite-to-martensite did
not occur homogeneously in their specimens. Further investigations by Shaw and
Kyriakides [2] on Ti-Ni sheets also showed that the phase transformation occurred
by the nucleation and propagation of fronts, and that because of the exothermic
and endothermic nature of the austenite-to-martensite and martensite-to-austenite
transformations, respectively, there were substantial measurable temperature changes
in the gage section of their sheet tensile specimens. These temperature changes also
result in an "apparent hardening" of the nominal superelastic stress-strain curves, as
observed by Entemeyer et al. (2000).
With this brief introduction, the purpose of this thesis is to formulate and numerically implement a crystal-mechanics-based constitutive model for shape-memory
materials, and to verify whether the model is able to capture the major features of
the experimentally-measured effects of crystallographic texture on superelasticity of
polycrystalline Ti-Ni alloy, and also to numerically capture the inhomogeneous deformation associated with the nucleation and propagation of transformation fronts,
and also the apparent hardening of the nominal stress-strain curves observed in the
experiments as the loading rate is increased.
The plan of this thesis is as follows:
In Section 2.1, we formulate a rate-
independent single-crystal constitutive model, where the inelastic deformations occur
by phase transformations. We have implemented our constitutive model in the finiteelement program ABAQUS/Explicit (1999,2001); algorithmic details of the timeintegration procedure used to implement the model in the finite-element code are
given in Appendix A. This computational capability allows us to perform two types
of finite-element calculations: (i) where a finite-element represents a single grain
and the constitutive response is given through a single-crystal constitutive model,
and (ii) where a finite-element quadrature point represents a material point in a
polycrystalline sample and the constitutive response is given through a Taylor-type
polycrystal model.
In Section 2.2 the results from experimental measurements of crystallographic
texture of a Ti-Ni rod-alloy are described.
20
Material parameters appearing in the
constitutive model for this alloy have been evaluated. The procedure to determine
these parameters from differential scanning calorimetric (DSC) measurements, and a
room temperature isothermal superelasticity experiment in tension are also outlined in
Section 1-3. We show that our model is able to reproduce the stress-strain curve of the
initially-textured Ti-Ni alloy in simple tension. Next, with the model so calibrated,
we show that the predictions for the stress-strain curves from the model are in good
agreement with superelasticity experiments on the same pre-textured material in
simple compression, thin-walled tubular torsion, combined tension-torsion, and path
change tension-torsion.
To determine the degree to which the initial texture controls the macroscopic
asymmetry in the superelastic response in tension and compression, we have also
compared the predicted stress-strain responses in simple tension and simple compression if the initial texture is taken to be random. Our calculations show that in this
case the response in compression is very similar to that in tension. Accordingly, we
conclude that crystallographic texture is the prime cause for the observed tensioncompression asymmetry in shape-memory materials.
In Section 2.2.1, we evaluate the applicability of a Taylor-type model for inelastic
deformations by phase transformation. Some previous work done by Bhattacharya
and Kohn (1996,1997) shows that the Taylor model provides a good approximation in
estimating the transformation induced strain. Our calculations also show that such a
Taylor-type model is also able to predict reasonably well the macroscopic stress-strain
curves.
In Section 2.2.2 we examine the applicability of our model to another Ti-Ni alloy for which Shaw and Kyriakides (1995) have conducted careful experiments at a
variety of different temperatures. Unfortunately, these authors do not report on the
initial crystallographic texture of their material. However, since they conducted their
experiments on drawn Ti-Ni wires, we assume that their material has a texture similar to our drawn rods. We estimate the constitutive parameters for their material
from their DSC results, and their stress-strain results from a pseudoleastic tension
test at representative temperature. We show that the predictions for the stress-strain
21
response from our constitutive model are in good agreement with the results from
their superelastic experiments at a few temperatures, 0 > Oaf.
Shaw and Kyriakides (1995) also report on displacement controlled experiments
at temperatures in the range 0 ms < 0 < Oaf. In these experiments the martensite that
forms during forward deformation does not completely transform back to austenite
upon reversing the deformation and decreasing the stress to zero. Although Shaw
and Kyriakides (1995) did not subsequently increase the temperature at zero stress
to 0 > 0as to show the shape-memory effect, we have numerically simulated such an
experiment, and show that our model is able to also capture the shape-memory effect
by transformation.
In Section 2.3 we investigate the superelastic response of initially-textured Ti-Ni
sheet. Here we show the constitutive model, when suitably numerically implemented
and calibrated, is shown to accurately predict the anisotropic superelastic response of
tension specimens which were cut along different directions to the rolling direction of
the sheet. Also, the strain-temperature cycling experiments under different constant
axial stresses are predicted with reasonable accuracy. The effects of self-heating and
cooling due to the exothermic and endothermic nature of the austenite-to-martensite
and martensite-to-austenite transformations were investigated by performing superelastic tension experiments at strain rates which are high enough to result in nonisothermal testing conditions. The thermo-mechanically coupled theory is able to
capture the resulting inhomogeneous deformation associated with the nucleation and
propagation of transformation fronts, and also the "apparent hardening" of the nominal stress-strain curves observed in the experiments.
In Section 2 we have successfully applied our crystal-mechanics-based theory to
model the superelastic behavior of Ti-Ni rods and sheets under various multi-axial and
thermo-mechanical loading conditions. However, as expected, the three-dimensional
crystal-mechanics-based theory for polycrystalline materials is computationally intensive, and at present not ideally suited for application in routine engineering design.
Guided by the success of the crystal-mechanics theory, we formulate a simpler
"isotropic" theory in Chapter 3, in which the major difference is that the evolution
22
equation for FP is now given by
FPFP-'=
where
NP,
(1.1)
denotes a single transformation rate, and NP a suitable symmetric trans-
formation tensor which determines the direction of transformation and its magnitude. Unlike the crystal-mechanics theory in which the transformation tensors are
determined by crystallographically based dyads S', the transformation tensor in the
isotropic theory is based on considerations of the stress state during transformation.
Although a crystal-mechanics theory is more accurate, an isotropic theory based on
such a flow rule, when suitably numerically implemented and calibrated, and its range
of applicability verified, should be more immediately useful for the design of components and systems made from shape-memory materials.
The plan of Chapter 3 paper is as follows.
We develop the isotropic model
in Section 3.1; we have implemented the model in the finite element programs
ABAQUS/Explicit and ABAQUS/Standard (2001). Algorithmic details of the timeintegration procedure and Jacobian matrices for the numerical implementation are
given in Appendix E and F, respectively. In Section 3.2.1 and 3.2.2 we evaluate
the applicability of this isotropic model to reproduce the experimentally-measured
superelastic response of the polycrystalline Ti-Ni sheets and rods, which experimental results are reported in Chapter 2. The two manifestations of superelasticity stress-strain response at a fixed temperature and strain-temperature response at a
fixed stress -
are explored. In Section 3.3 we use our computational capability to
show that it may be used to analyze the response of a technologically important
structure -
a bio-medical stent.
We conclude and provide suggestions for future work in Chapter 4.
23
(a)
(b)
Figure 1-1: (a) Shape-memory alloy-based actuation device (Baron et al., 1994). (b)
Scanning Electron Microscope (SEM) image of a Scimedt" artery stent (Serruys,
1997).
24
FIXED 0
FT
.d
z
C,,
d:
A-M
CO)
Cf)
M-A
If
TIME [s]
Input
STRAIN
Output
(a)
FIXED a
W
z
CO)
CL
w
0W
A-MM-
TIME [s]
Input
TEMPERATURE [K]
(b)
Output
Figure 1-2: Schematic diagram of the (a) superelastic, and (b) strain-temperature
cycling response. A-M denotes the austenite-to-martensite transformation whereas
M-A denotes the martensite-to-austenite transformation.
25
10001
1
1
900
1
1
-
TENSION
- -
COMPRESSION
1
800
700
-
-
600
0.
o
Cl)
Cl)
500
400
--
-
300
200
100
C
0.01
0.02
0.04
0.03
STRAIN
0.05
0.06
0.07
Figure 1-3: Experimental stress-strain curves for Ti-Ni rod at 6 = 298 K in simple
tension and simple compression. For compression the absolute values of stress and
strain are plotted.
26
550
.
I
500
0 EXPERIMENT: ROLLING
A EXPERIMENT: TRANSVERSE
450 400 -
-
cz 350 -
g)
300 -
U
250 -
w
200 -150 100 50
0O
0
A
0.01
0.02
0.03
STRAIN
0.04
0.05
0.06
Figure 1-4: Experimental stress-strain curves for Ti-Ni sheet at 0 = 300 K in simple
tension along the rolling and transverse direction
27
600
o EXPERIMENT: 0.00003 /s
A EXPERIMENT: 0.00084 /s
o EXPERIM ENT : 0.002 /s
500
-
111
1
0
400 [
0
03
30
00
CL)
w
300 1
I
I
I
30
0 o
I-I
13
Cl)
200
100
3013
0
0
0.01
0.02
0.03
0.04
0.05
STRAIN
0.06
0.07
0.08
0.09
Figure 1-5: Experimental stress-strain curves for a Ti-Ni sheet at 0 = 300 K in simple
tension at strain rates of 0.00003/sec, 0.00084/sec, and 0.002/sec.
28
Chapter 2
Crystal-mechanics-based
constitutive model
2.1
Single-crystal constitutive equations
In this section we summarize the three-dimensional crystal-mechanics based constitutive model of Thamburaja and Anand (2001,2002), which has recently been reformulated by Anand and Gurtin (2002) to account for thermal influences within a rigorous
thermodynamic framework.1
The overall inelastic deformation of a crystal is always inhomogeneous at length
scales associated with the fine-scale microstructures accompanying the austenitemartensite phase transformations. Thus, for the continuum level of interest here,
the inelastic deformation should be defined as an average over a volume element that
must contain enough transformed regions to result in an acceptably smooth process.
The such smallest volume element above which the inelastic response may be considered smooth, is labelled a representative-volume element (RVE). In our constitutive
model we choose a single-crystal as a representative volume element. The model
does not explicitly account for the fine-scale microstructures, but only for the volume
fractions of various types of fine structure. Thus the spatially continuous fields that
'This model is not intended to describe a material initially in the martensitic state and inelastic
deformation caused by reorientation and de-twinning of the martensitic variants.
29
define the theory are to be considered as averages meant to apply at length scales
which are large compared to those associated with the fine structure.
Using the standard notation of modern continuum mechanics,2 we recall that
the deformation gradient F maps referential segments dX to segments dx = FdX
in the deformed configuration. We base the theory on the following multiplicative
decomposition of the deformation gradient: 3 F = FeFP. Here (i) FP(X) represents
the local deformation of referential segments dX to segments dl = FP(X)dX in
the relaxed lattice configuration due to the generation, growth and annihilation of
austenitic/martensitic fine structure in a microscopic neighborhood of X. (ii) FI(X)
represents the mapping of segments dl in the relaxed lattice configuration into segments dx = F (X)dl in the deformed configuration due the "elastic mechanisms" of
stretching and rotation of the lattice.
In the equilibrium theory of austenite-martensite phase transitions the strain
energy minimizers (in the sense of minimizing sequences) generally appear in the
form of fine structure, involving abrupt transitions between austenite and twinned
martensite in which the transitions are jumps in the deformation gradient of the
form So = bo 0 mo, with bo a vector whose magnitude represents the transition
strain, and mo a unit vector normal to the habit plane (the plane of the transition).
A result of the equilibrium theory is a catalog of such rank-one tensors for various
shape-memory alloys (cf. James and Hane (2000) and the references therein). Here
we take, as a basic ingredient of the theory, a system of N such transformation tensors S' = b Om', i = 1,2, ...
, N,
where b are vectors whose magnitudes represent
the transformation strains, and mi are unit vectors representing the corresponding
2
Notation: V and Div denote the gradient and divergence with respect to the material point
X in the reference configuration; grad and div denote these operators with respect to the point
x = y(X, t) in the deformed configuration, where y(X, t) is the motion; a superposed dot denotes
the material time-derivative. We write sym A, skw A, respectively, for the symmetric, and skew
parts of a tensor A. Also, the inner product of tensors A and B is denoted by A . B, and the
magnitude of A by JAl = VA-A. Here repeated indices do not imply a summation over all the
possible values the indices can take. For summation the E sign is used.
3
This decomposition is identical in form to the classical decomposition used to formulate dislocation based finite single-crystal plasticity. In this work we use the superscript p for two reasons: (i)
to stress similarities between the present theory and single-crystal plasticity; and (ii) as shorthand
for "phase transformation".
30
habit-plane normals; these constant vectors are not necessarily orthogonal. We denote by $'(X, t), 0 < ' < 1, the local volume fraction of martensite at X associated
with the ith system at time t. The presumption that the phase transformations take
place through nucleation and growth of plate-like volume elements is based on the
hypothesis that the evolution of FP is governed by the transformationrates d via the
relation FP = LPFP, LP = E
§ . For system i, transformationfrom austenite to
martensite, abbreviated a -+ m, occurs when d' > 0; a transformationfrom martensite to austenite, abbreviated m -+ a, occurs when d < 0. We write
=
for
the (local) total volume fraction of transformed regions, and assume, as is natural,
that 0 <
(t) K 1.
The underlying constitutive equations relate the following basic variables :4
free energy density per unit reference volume,
F,
det F > 0,
deformation gradient,
9,
6 > 0,
absolute temperature,
T,
T = T T,
Cauchy stress,
FP,
transformation deformation gradient,
(b', mb),
transformation systems,
Fe = FFP~ 1,
0 < C < 1,
martensite volume fraction for the ith system,
0 <
total martensite volume fraction,
det F' > 0,
Ce = FeTFe,
Ee = (1/2) (Ce
1,
elastic deformation gradient,
elastic right Cauchy-Green tensor,
-
1),
Te = (det F)Fe-TFe~T,
elastic strain,
elastic stress.
The constitutive equations are:
4
Using the notation of modern continuum mechanics, we write F'etc. throughout.
31
1
= (Fe)-l, FP-T
=
(FP)
T,
1. Free energy: The free energy is taken in the separable form5
O(Ee )7)
,e((Ee,
0, )
=
(1/2)Ee - C( )[Ee]
0'9(0) = c(O - Oo) - c
,tr
(6
(6) +4,tr(O,),
+4
= Oe (Ee,O,)
-
9 0)A(() C( )[Ee],
(2.1)
(2.2)
(2.3)
(2.4)
.
Here C( ) is the elasticity tensor and A()
6
(0
In (O/ 6 o),
) = (AT/OT) (0 - OT)
the reference temperature
-
with
is the thermal expansion tensor at
0. The constant c is the specific heat per unit vol-
ume of the reference configuration. The parameter
OT
is the phase equilibrium
temperature, AT is the latent heat per unit reference volume of the a
+ m
transformation at temperature OT. 6
2. Stress:
The elastic stress is given by
Te
Ee
=C()[El - A( )(O - Oo)].
(2.5)
We define the resolved force on the ith system by
=
b. (Ce Te)mi.
(2.6)
Note that since the magnitude of b' represents a transformation strain, T' has
units of energy per unit reference volume, and it is for this reason we call it a
resolved force rather than a resolved stress. Further, the derivative
b
(AT/OT)(0 - OT),
=
(2.7)
represents an energetic back force,7 and the driving force for phase transforma-
tion on the ith system is given by
fi
= (i
-
b).
(2.8)
,3where the scalar constants g'J = g3' characterize ener'A term of the form (1/2)>gi'
getic interactions between the transformation systems, may be included in the free energy ptr, but
we shall neglect such a contribution in our application of the theory to Ti-Ni.
6
Here we are guided by the one-dimensional development of Abeyaratne and Knowles (1993). An
(0 - Oir) (%.
obvious generalization of ?p' would be to replace it by
'We use this terminology by analogy to the "back stress" in the crystal plasticity literature.
Z'(r/ar)
32
From the second law of thermodynamics, the dissipation accompanying the
phase transformation is given by
Zfili > 0.
(2.9)
From the inequality (2.9), we obtain for the individual transformation systems
that
(2.10)
fi i > 0.
3. Transformation conditions:
In the rate-independent theory developed here, transformation is assumed to
be possible if the driving force fi reaches a critical value. From the inequality
(2.10) the transformation conditions are
f, for
fi
f, >
(2.12)
<0.
fi =-f, for
Here
(2.11)
> 0,
0 is the constant transformationresistance for the ith system. In the
rate-independent limit the bounds8 on the driving force for phase transformation
are
-f < ft < fcz.
(2.13)
4. Flow rule:
FP={
%'
b9
rm
FP,
with martensitic volume fractions consistent with 0 <
< 1 and 0 <
< 1.
Moreover,
(i) if -fe < fi < fc, then
(ii) if fi = fj,
' = 0;
0 < i <1 and 0
); 0
8
< 1, then
and
(fi -
This is analogous to the yield function in crystal-plasticity.
33
fi)
< 0;
(2.14)
(iii) if f i
0<$ <1 and 0<
= -fP,
<- 0 and
(iv) if fi
fR and
=
(v) if fi =f
and
( = 1, or if fi =
< 1, then
(f±i + fj) ) 0;
fi
and ( = 0, then (
(2.15)
0;
=1, then d=0.
The consistency conditions (2.14) and (2.15) serve to determine the transformation rates (.
5. Entropy; energy balance; heat flux:
The entropy 17 is given by
17
O
-
- c ln (O/6o) + Ee - C( )[A( )] - (AT/T) .
(2.16)
The balance of energy may be written as
9?)
=
-Divq + Zfi
+r,
(2.17)
where r is the heat supply per unit volume of the reference configuration, and
the referential heat flux q is taken to be governed by Fourier's law
q = -K(
)VO,
(2.18)
where K(s) is the thermal conductivity tensor at 00. Using (2.16) and (2.18),
the energy balance (2.17) becomes
c
=
Div {K( )V6} + (AT/OT) 0 + E fi'i - 9Ee - C( )[A( )] + r,
(2.19)
To complete the theory for a particular material the constitutive parameter/functions that need be specified are
C( ), A( ),7b', mi, OT, AT, f,, c , K((
Finally, the standard partial differential equation for the stress referred to the
deformed configuration is
div T + J-f = 0,
34
(2.20)
where div represents the divergence operator in the deformed configuration, J =
det F, and f is the body force per unit volume in the reference configuration (the
body force f is assumed to include inertial forces). The differential equation for the
temperature referred to the deformed configuration is
J-1 c = div
{
J-FK( )F Tgrad 0} + J-
a
(AT/OT) 0
-
+ J'
fi
J'OE -C( )[A( )] + J'r.
(2.21)
The numerical simulations are carried out using the finite-element computer program ABAQUS/Explicit (1999,2001), for which we have written a user material subroutine to implement our constitutive model. Algorithmic details of the numerical
implementation are given in Appendix A.
35
2.2
Evaluation of the constitutive model for a
polycrystalline Ti-Ni alloy : Rod-texture
Suitably thermo-mechanically processed and heat-treated Ti-Ni at 55.96 wt.% Ti
drawn rods of 12.70mm diameter, intended for superelastic applications, were obtained from a commercial source. Experimental measurements of crystallographic
texture of the as-received Ti-Ni were carried out by X-ray irradiation using a Rigaku
RU 200 diffractometer. Pole figures were obtained by using the Schultz reflection
method with copper-K radiation. To process the experimental data, the PoPLa software package (Kallend et al., 1989) was employed. Each measured pole figure was
corrected for background and defocusing, and also extrapolated for the outer 150.
The {111} , {100}, and {110} pole figures for the as-received Ti-Ni looking into the
rod-axis (X3) are shown in Figure 2-1. This figure also shows our numerical approximation of this measured texture using 729 and 343 unweighted grain orientations.
The texture representation using 729 grain orientations is slightly better that that
using 343 grain orientations. In most of the calculations reported below we shall use
the 729 grain orientation representation of the initial texture. However, we note that
the computed stress-strain curves using the set of 343 grain orientations are very close
to those using the 729 grain orientations, and the smaller number of orientations may
be used for reasons of computational efficiency.
By using differential scanning calorimetric (DSC) techniques, we have determined
the transformation temperatures for our Ti-Ni bar-stock, Figure 2-2. They are:
Oms = 251.3 K, Omf = 213.0 K, Oas = 260.3 K, Oaf = 268.5 K.
(2.22)
The volume fraction and temperature dependence of the anisotropic elastic constants of Ti-Ni are not well-documented in the literature. Here, for simplicity, the
values of the elastic constants are taken to follow the rule of mixtures
C( , 0) = (1 -
)Ca(o) + C m (9).
The elastic constants for the cubic austenite phase in Ti-Ni at room temperature are
taken as (Brill et al., 1991) :
36
C 1 =130 GPa,
C"2 = 98 GPa,
C44= 34 GPa.
We have been unable to find experimentally-measured values of the anisotropic elastic
moduli of single-crystal monoclinic martensite of Ti-Ni. Typically, the Young's Modulus of austenite is about two to three times that of martensite, while the Poisson's
ratios for the material in the two different phases are approximately equal to each
other. Guided by this information, we assume that anisotropic elastic constants of
the monoclinic martensite may be approximately treated as those of a cubic material,
and that the corresponding values of the stiffnesses, Cj, are one half as large as those
for the austenite:
Cm = 65 GPa,
Cm = 49 GPa,
C
=
17 GPa.
We realize that this description for the composite elastic constants is rather simplified
and approximate. However, since the main purpose of this paper is to model the
inelastic response characteristics of shape-memory materials, we shall not further
pursue the matter of a more refined description of the elastic moduli of a two-phase
austenite-martensite mixture.
For cubic materials, the thermal expansion tensor is isotropic, A a = aal, with aa
denoting the coefficient of thermal expansion for the austenite phase. The thermal
expansion tensor for the martensite phase is also assumed to be isotropic i.e. Am
=
am 1, with am denoting the coefficient of thermal expansion for the martensite phase.
Here, the composite thermal expansion tensor is taken to follow the rule of mixtures
A = al = (1 -
)A a +Am,
with aa and am for Ti-Ni in the austenitic and martensitic conditions taken as (Boyd
and Lagoudas, 1996):
aa = 11 x 10- 6 /K,
a m = 6.6 x 10- 6 /K.
The crystallographic theory of martensitic transformation (CTM) shows that
phase transformation in Ti-Ni can occur on 192 possible transformation systems
37
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Table 1. Transformation Systems
m
m
m2
bZ
bZ
-0.8888 -0.4045 0.2153 0.0568 -0.0638
-0.8888 0.4045 0.2153
0.0568 0.0638
0.0991
-0.8888 0.2153 -0.4045 0.0568
-0.8888 0.2153 0.4045
0.0568 0.0991
-0.8888 -0.2153 0.4045 0.0568 -0.0991
-0.8888 -0.2153 -0.4045 0.0568 -0.0991
0.0638
-0.8888 0.4045 -0.2153 0.0568
-0.8888 -0.4045 -0.2153 0.0568 -0.0638
0.0638 0.0568
0.4045 -0.8888 0.2153
-0.4045 -0.8888 0.2153 -0.0638 0.0568
0.2153 -0.8888 -0.4045 0.0991 0.0568
0.0568
0.2153 -0.8888 0.4045 0.0991
-0.2153 -0.8888 0.4045 -0.0991 0.0568
-0.2153 -0.8888 -0.4045 -0.0991 0.0568
0.0568
0.4045 -0.8888 -0.2153 0.0638
-0.4045 -0.8888 -0.2153 -0.0638 0.0568
0.2153 0.4045 -0.8888 0.0991
0.0638
0.2153 -0.4045 -0.8888 0.0991 -0.0638
0.4045 0.2153 -0.8888 0.0638
0.0991
-0.4045 0.2153 -0.8888 -0.0638 0.0991
0.4045 -0.2153 -0.8888 0.0638 -0.0991
-0.4045 -0.2153 -0.8888 -0.0638 -0.0991
-0.2153 0.4045 -0.8888 -0.0991 0.0638
-0.2153 -0.4045 -0.8888 -0.0991 -0.0638
b,3
0.0991
0.0991
-0.0638
0.0638
0.0638
-0.0638
-0.0991
-0.0991
0.0991
0.0991
-0.0638
0.0638
0.0638
-0.0638
-0.0991
-0.0991
0.0568
0.0568
0.0568
0.0568
0.0568
0.0568
0.0568
0.0568
(Hane and Shield, 1999). The CTM theory is discussed and summarized in Appendix
B and C.
However, it is not clear whether all the 192 possible systems are actually operative
during thermo-mechanical loading. Here, we shall only consider the 24 transformation
systems which are unambiguously observed in experiments and used by a variety
of recent researchers (e.g. Matsumoto et al., 1987; Lu and Weng, 1998; Gall and
Sehitoglu, 1999). The components of the 24 transformation systems (mi, b') with
respect to an orthonormal basis associated with the parent cubic austenite crystal
lattice are given in Table 1.
Recall that the driving force on each transformation system is taken to be given
38
by
fi
=
(AT/OT) (0 -
r-
OT).
Using the measured phase transformation temperatures (Figure 2-2), the value
for the phase equilibrium temperature,
OT
is given by
OT,
= (1/2) {Oms +
256 K.
Oas}
Further we assume that the the critical values of the driving force f, for the 24
systems are all equal and constant:
f=
constant.
In principle, the values of the latent heat for phase transformation, AT, and the
critical value of the driving force,
f,,
should be determined from experiments per-
formed on single crystals of Ti-Ni. However, such single crystals are difficult to grow,
and we did not have access to single crystals of this material. Instead, the values of AT
and f, are estimated from experiments on polycrystalline Ti-Ni as follows. Consider
an idealized schematic stress-strain curve for a superelastic tensile test on a polycrystalline shape-memory material, Figure 2-3. Let o-am denote the value of the stress
at which martensite nucleates from austenite during the forward transformation, and
let oma denote the stress level at which austenite nucleates from martensite during
the reverse transformation. Following the analysis performed by Knowles (1999), the
value of the mean stress, defined as
0-o
=
1
(0-am + Oma) ,
is given by
O-o = (AP0'Y/0T ET) (0 - OT),
where AP2'Y is the one-dimensional polycrystalline counterpart of the parameter AT and
ET is the macroscopic transformation strain. Thus, if one knows
DSC measurements, and one estimates
0-am, 0ma
0
T
from independent
and ET from a superelastic tensile
test at a temperature 0, then the macroscopic value of the latent heat for phase
transformation, Apol,
for a polycrystalline material is easily estimated. As shown in
39
Knowles (1999), the critical value for the macroscopic driving force, fP1OY , is then
given by
fI'ly =
CT (am
-
0),
where ET is the value of the transformation strain in tension for a polycrystalline
material, see Figure 2-3.
Estimates of the values of the corresponding quantities (AT,
fc)
at the single crystal
level are then obtained from
AT ~ A' 1 ",
and
C~ frO_Y
Figure 1-3 shows the stress-strain curve from a superelastic tension test performed
on our initially-textured Ti-Ni at 298 K. The geometry of the tension-compression
specimens is shown in Figure 2-4. The experiment was conducted at a very low
constant true strain rate in order to ensure isothermal testing conditions. An extensometer was used to obtain the macroscopic strain in the gage section of the specimen.
The experimental stress strain curve shows some residual deformation after unloading. This is invariably observed in most experiments on commercial polycrystalline
SMAs; seldom is there complete recovery. From this figure,
ram
~ 470 MPa,
Using, the value O
=
~ 170 MPa,
ama
ET
~ 0.054.
256K, estimated from our DSC experiments, we obtain
fPOY = 8.2 MJ/m 3
Agl =100 MJ/m 3 ,
for the polycrystalline material. Hence, the estimates of the values of the corresponding quantities at the single crystal level are
AT
~ 100 MJ/m 3 ,
and
f, ~~8.2 MJ/m 3 .
In our finite-element model of a polycrystalline aggregate, each finite-element
represents one crystal, and a set of crystal orientations which approximate the initial
crystallographic texture of the shape-memory alloy are assigned to the elements.
The macroscopic stress-strain responses are calculated as volume averages over the
40
entire aggregate. Calculations for simple tension on an aggregate of 729 unweighted
different grain orientations representing a polycrystalline material with the initial
crystallographic texture, Figure 2-1, were carried out using these estimates for the
material parameters. Figure 2-5a shows the initial finite-element mesh. The finiteelement mesh after 6% deformation in tension is shown in Figure 2-5b, together
with the contours of the martensite volume fraction. The quality of the curve-fit is
shown in Figure 2-5c. The numerically-computed stress-strain response is close to the
experimentally-observed one. Given the number of approximative assumptions used
to arrive at this curve-fit, the result is very encouraging.
Using these values of the material parameters and the numerical representation
of the measured initial texture, Figure 2-1, we have also carried out numerical simulations of simple compression and thin-walled tubular torsion, and compared the
calculated stress-strain curves against corresponding physical experiments at 298 K.
The initial finite-element mesh for the simple compression simulation is shown in
Figure 2-6a. The finite-element mesh after 5 % compression is shown in Figure 2-6b,
along with the contours of the martensite volume fraction. The prediction of the
stress-strain curve from the constitutive model is shown in Figure 2-6c, where it is
compared against corresponding experimental measurements. The experimentallymeasured superelastic stress-strain response is well-approximated by the predictions
from the constitutive model.
To demonstrate the numerically-predicted tension-compression asymmetry, we
plot the numerical stress-strain curves in tension and compression in Figure 2-7a.
On comparing the curves in this figure with the corresponding experimental curves,
Figure 1-3, we conclude that the constitutive model captures the following major
features of the observed tension-compression asymmetry rather well:
" The stress level required to nucleate the martensitic phase from the parent
austenitic phase is higher in compression than in tension.
" The transformation strain measured in compression is smaller than that in
tension.
41
* The hysteresis loop generated in compression is wider (measured along the stress
axis) than the hysteresis loop generated in tension.
Figure 2-1 shows that the Ti-Ni bar has a strong {111} texture. Using our constitutive model and the estimated single crystal material parameters, we have calculated the stress-strain response for a single crystal subjected to tension and compression along its [111]-direction. The calculated stress-strain response is plotted in
Figure 2-7b. Comparison of Figure 2-7a with Figure 2-7b clearly shows that the
tension-compression asymmetry in the polycrystalline specimen has its origins in the
crystallographic texture of the as-received Ti-Ni bar.
To confirm this conclusion we have repeated the tension and compression simulations for a polycrystalline specimen using a set of 729 crystal orientations representing
a "random" texture, instead of the actual crystallographic texture in the Ti-Ni bar
shown in Figure 2-1. The {111} pole figure corresponding to this random initial texture is shown in Figure 2-8a. All other material parameters used in these simulations
were the same as those used in the previous calculations. The predicted tension and
compression superelastic stress-strain curves using the random texture are shown in
Figure 2-8b. This result shows that in comparison to the result from the calculation
using the actual initial texture, Figure 2-7a, there is not much asymmetry between
the compression and tension curves. Also, the small asymmetry in the stress levels
between tension and compression observed in the calculation using the random initial texture is in the reverse order. The experiments, Figure 1-3, and the numerical
simulations using the actual initial texture for the rod, Figure 2-7a, show that the
compression curves are higher than those for tension, whereas the calculation using
the random texture shows the reverse trend. Thus, we conclude that crystallographic
texture is the prime cause for the observed tension-compression asymmetry in shapememory alloys.
Finally, the specimen geometry for the thin-walled tubular torsion experiment is
shown in Figure 2-9. The initial mesh used in the simulation for tubular torsion is
shown in Figure 2-10a. The deformed mesh after a shear strain of 9% is shown in
Figure 2-10b, together with the contours for the martensite volume fraction. The
42
predicted nominal shear stress-strain curve is shown in Figure 2-10c, where it is
compared against the corresponding experimentally-measured curve from a tubular
torsion experiment performed on a thin-walled specimen under a very low shearing
strain rate at 298 K. The prediction is in good accord with the experiment.
We have also evaluated the predictive capability of our model for (i) a proportionalloading, tension-torsion experiment, and (ii) a path-change, tension-torsion experiment. These experiments were also performed under isothermal testing conditions at
room temperature (0 = 298 K).
Proportional-loading tension-torsion experiment:
The strain-history imposed on a thin-walled tubular specimen during the proportionalloading tension-torsion experiment is shown in Figure 2-11a. The axial strain E is
increased at a constant rate from 0% to 3% in 500 seconds, while the shear strain -y is
increased at a constant rate from 0% to 4.5% in the same time period. These strains
are then linearly reduced to zero, in another 500 seconds.
The initial finite-element mesh using an aggregate of 768 unweighted crystal orientations for the combined tension-torsion simulation is shown in Figure 2-11b, and
the finite-element mesh after 3% strain in tension and 4.5% strain in shear is shown
in Figure 2-11c, along with the contours of the martensite volume fraction.
Predictions of the axial-stress versus axial-strain, and shear-stress versus shearstrain from the constitutive model are compared against corresponding experimental
measurements in Figures 2-11d and 2-11e, respectively. The measured superelastic
stress-strain response in the tension-torsion experiment is well-approximated by the
predictions from the constitutive model.
Path-change tension-torsion experiment:
The strain-history imposed on a thin-walled tubular specimen during the pathchange tension-torsion experiment is shown in Figure 2-12a. First, the axial strain is
increased at a constant rate from 0% to 3% in 150 seconds, while the shear strain is
maintained at 0%. After this, the shear strain is linearly increased from 0% to 7%
in 350 seconds while the tensile strain is maintained at 3%. Next, the shear strain
43
is linearly reduced from 7% to 0% in 350 seconds, while still maintaining the tensile
strain at 3%. Finally, the tensile strain is linearly reduced from 3% to 0% in another
150 seconds, while maintaining the shear strain at 0%.
The initial finite-element mesh using an aggregate of 768 unweighted crystal orientations for the path-change tension-torsion simulation is shown in Figure 2-12b, and
the finite-element mesh after 3% strain in tension and 7% strain in shear is shown in
Figure 2-12c, along with the contours of the martensite volume fraction.
Predictions of the axial-stress versus axial-strain, and shear-stress versus shearstrain from the constitutive model are compared against corresponding experimental
measurements in Figure 2-12d and 2-12e, respectively. Again, the measured superelastic stress-strain response in the tension-torsion experiment is well-approximated
by the predictions from the constitutive model.
Of particular note is the result that when the axial strain reaches 3% and is
maintained at this level, the axial stress drops when the specimen is subsequently
sheared. The axial stress rises again when the direction of shearing is reversed. As
shown by the arrows in Figure 2-12d, this response is remarkably well-captured by
the simulation.
Further, note from Figure 2-12e, that the shear stress has a negative value after
the shear strain has completed its cycle back to zero. This shear stress decreases back
to zero only when the axial strain is finally reduced back to zero. This response is
also well-predicted by the numerical simulation.
44
Taylor model
2.2.1
For polycrystalline materials, a widely-used averaging scheme is based on the assumption that the local deformation in each crystal is homogeneous and identical to the
macroscopic deformation gradient at the continuum material point (Taylor, 1938).
The compatibility between crystals is automatically satisfied in this approximation;
however, equilibrium holds only inside a crystal, but is violated across crystal boundaries. For such a model, with T(k) denoting the constant Cauchy stress in each crystal,
the volume-averaged Cauchy stress is given by
N
k=1
where
v(k)
is the volume fraction of each crystal in a representative volume element.
When all crystals are assumed to be of equal volume, the stress T is just the number
average over all the crystals:
N
k=1
Taylor model simulations in simple tension, simple compression, simple shear and
path-change tension-torsion were performed using a single ABAQUS-C3D8R element,
and using 729 unweighted grain orientations to represent the initial rod-texture, Figure 2-1. The material parameters used in the Taylor model simulations were the same
as those calibrated for the full finite-element simulations of the polycrystal discussed
in the previous section. Figures 2-13a, 2-13b and 2-13c compares the stress-strain
predictions from the Taylor model against the actual experiments, as well as the
full finite-element calculations. Figure 2-14 compares the stress-strain predictions
from the Taylor model against the actual path-change tension-torsion data, as well
as predictions from the full finite-element calculations. The Taylor model compares
very well to the full finite-element calculations, and therefore it provides a reasonably
accurate and computationally-inexpensive method for determining the response of
textured Ti-Ni in a multi-axial setting and moderately complex loading modes.
45
2.2.2
Effect of temperature on the deformation of polycrystalline Ti-Ni
In this section we examine the applicablity of our model to another Ti-Ni alloy for
which Shaw and Kyriakides (1995) have conducted careful isothermal, low strain rate
experiments at a variety of different temperatures in the range 255.6 K < 0 < 373.2 K.
Unfortunately, these authors do not report on the initial crystallographic texture of
their material. However, since they conducted their experiments on drawn Ti-Ni wires
with a processing history similar to that of our own material, we assume that their
material has a texture which may be approximated' by the numerical representations
of the texture of our drawn rods, Figure 2-1.
We estimate the constitutive parameters for their material from their DSC results,
and their stress-strain results from a superelastic tension test at representative temperature. The DSC measurements of Shaw and Kyriakides (1995) (their, Figure 1)
yield the following values for the transformation temperatures:
6ms = 272.2 K,
9
mf
= 203.2 K, 0a, = 302.7 K, Oaf = 335.2 K.
(2.25)
The material parameters for their Ti-Ni wire were calibrated by fitting the constitutive
model to the superelastic tension experiment conducted at 343.2 K using the methodology outlined in Section 2.2. The quality of the curve-fit is shown in Figure 2-15b.
The numerical calculations shown in this figure correspond to using the full finite element model with 729 elements representing 729 grain orientations, as well as a corresponding single element Taylor model calculation. The full finite element model of the
polycrystal was used to estimate the material parameters; the numerically-computed
stress-strain response from this calculation is close to the experimentally-observed
one.
The thermo-elastic constants used in our calculations are
Elastic moduli for austenite: Ca = 130 GPa, Ca
=
98 GPa, CA4
Elastic moduli for martensite: Cm
=
49 GPa, C4
9
=
65 GPa, Cm
=
21 GPa;
=
10.5 GPa;
Wire texture is expected to show a sharper {111} component than the rod texture.
46
Coefficients of thermal expansion: aa = 11 x 10- 6 /K, a m = 6.6 x 10
6 /K.
The single crystal material parameters used to obtain this fit for the polycrystalline
Ti-Ni wire of Shaw and Kyriakides are :
Phase equilibrium temperature: OT
Latent heat:
AT =
=
287 K;
112 MJ/m3 ;
Critical transformation resistance: f
=
7.7 MJ/m 3 .
Figure 2-15a and 2-15c show the response predicted from the constitutive model compared against the tension experiments at two different temperatures: 333.2 K, and 353.2K.
The experimentally measured temperature variation of the superelastic stress-strain
response is well-predicted by the constitutive model.
Shaw and Kyriakides (1995) also report on displacement controlled experiments
at temperatures in the range 0,
< 0 < Oaf. In these experiments the martensite that
forms during forward deformation does not completely transform back to austenite
upon reversing the deformation and decreasing the stress to zero.
Although Shaw
and Kyriakides (1995) did not subsequently increase the temperature at zero stress
to show the austenite-martensite-austenite shape-memory effect, we have numerically
simulated such an experiment. In our simulation we employ the Taylor model using
a single ABAQUS-C3D8R element with 343 grain orientations to represent the initial
texture. The calculation was performed by first imposing an isothermal, 0 = 303.2 K,
strain controlled tension to 5% tensile strain, and then reversing the deformation to
reach a state of zero stress. As shown in Figure 2-16, the numerical prediction from
the model for this part of the simulation is close to the experimental measurements
of Shaw and Kyriakides (1995). The temperature in the simulation was then linearly
ramped up from 303.2 K to 305.8 K by imposing the temperature ramp on the nodes
of the finite element. Figure 1-15 shows that the model is able to capture the shapememory effect by transformation; it predicts full recovery to the austenite phase after
the temperature is increased to 305.8 K.
47
2.3
Evaluation of the constitutive model for a
polycrystalline Ti-Ni alloy : Sheet-texture
Suitably processed Ti-Ni sheet, 0.53 mm thick, intended for superelastic response was
obtained from a commercial source. Experimental pole figure measurements of the
initially-textured sheet were obtained using standard X-ray diffraction techniques.
The {111}, {110} and {100} experimental pole figures are shown in Figure 2-17a. A
numerical representation of the experimental pole figures using 48 weighted crystal
orientations was obtained by using the computer program PoPLa (Kallend et al.,
1989), Figure 2-17b. 10
The geometry of the tensile specimen used in the superelastic experiments is shown
in Figure 2-18a. Tensile specimens were cut for testing along three different directions: 00, 450 and 90' to the rolling direction. The finite element mesh comprising of
446 three-dimensional ABAQUS-C3D8R elements used in the numerical calculations
discussed below is shown in Figure 2-18b. Although not visible, the mesh has a very
slight taper such that the gage width at one end of the gage section is slightly narrower than that of the other end. This geometric "imperfection" in the finite-element
mesh is introduced to provide an initiation site for phase transformation. We have
previously shown that use of the Taylor-model (1938) for simulating the response of
polycrystalline Ti-Ni rods produces results which are in reasonably good agreement
with experiments, as well as full finite element calculations which do not make this
assumption. Accordingly, for computational efficiency, all numerical simulations for
polycrystalline Ti-Ni sheet reported in this section have been performed using the
Taylor-model (1938).
The thermo-elastic constants used in our calculations are
Elastic moduli for austenite: Ca = 130 GPa, C"2
98 GPa, CA4
=
21 GPa;
Elastic moduli for martensite: C' = 65 GPa, C'") = 49 GPa, C44= 10.5 GPa;
1
OThis small number of weighted grain orientations, which produces an acceptable representation
of the major components of the texture, is used for numerical efficiency. Use of a larger number of orientations does not significantly improve the texture representation, but increases the computational
time.
48
Coefficients of thermal expansion:
aa =
11 x 10- 6 /K, am = 6.6 x 10- 6 /K.
To obtain estimates of the phase transformation parameters a DSC measurement
needs to be conducted to obtain the phase equilibrium temperature,
OT.
Since there
was difficulty in obtaining pristine DSC experimental results for our polycrystalline
sheet Ti-Ni other methods of determining the material must be explored. Therefore an
estimate for the phase transformation parameters
{ T,
AT,
fc}
for Ti-Ni single crystals
is obtained by fitting the constitutive model to a set of superelastic experiments
conducted on our polycrystalline sheet at different temperatures.
The variation of the superelastic forward nucleation resolved force (Trm) and reverse nucleation resolved force
on an active transformation system i with tem-
(Tia)
perature can be plotted in a graph schematically shown in Figure 2-19. The forward
nucleation resolved force is the first instance where austenite to martensite transformation occurs. The reverse nucleation resolved force is the first instance where
martensite to austenite transformation occurs. From this, the intercept of the line
Tam
versus 0 at zero stress defines the martensite start temperature
is,
while the
intercept of the line~ri versus 0 at zero stress defines the austenite start temperature
0a,;
the mean value
1
T=
(2.26)
-(as + Oms)
2
determines the phase equilibrium temperature.
From (2.8) and (2.11) the forward transformation conditions give us
T'm(0) -
(2.27)
(AT/OT) (0 - OT) =f,
and the reverse transformation conditions (2.12) yield
Tma(0) -
(AT/OT) (0 - OT)
=
fc.
(2.28)
By adding (2.27) to (2.28), the mean values of these nucleation resolved forces
vary linearly with temperature i.e.
1
--(Tm (0) + Tra(0))
2
=
49
(AT/0T) (0 - OT).
The slope of the line
2(Taim
(0) + Tma(0)) versus 0 will give (AT/OT), which in turn
gives the latent heat of phase transformation, AT. The transformation resistance
fc
is then given by subtracting (2.28) from (2.27)
fc
=
(1/2) (Tam(0)
-
rTa()).
To evaluate these phase transformations parameters superelastic tension experiments were conducted on polycrystalline sheet specimens with the tension axis aligned
along the sheet rolling direction - the 0' orientation. The experiments were conducted
at three different temperatures, 289 K, 300 K and 306 K, under displacement control at
a very low nominal strain rate of 3 x 10-' /sec, to ensure (near) isothermal conditions.
The nominal stress-strain curves from these experiments are shown in Figures 2-20a,
2-20b and 2-20c, respectively". The corresponding numerical simulations at the three
different temperatures were carried out on the finite-element mesh shown in Figure 218b using 446 ABAQUS-C3D8R elements. For the tension simulation, one end of the
finite-element mesh is fixed in all its displacement degrees of freedom, while the other
end is subjected to a displacement history corresponding to a nominal axial strain
rate of ±3 x 10-5 /s.
Following the procedure outlined above the value of the fitted phase transformation parameters are :
Phase equilibrium temperature: 6
Latent heat: AT
=
=
271 K;
110 Mj/m3 ;
Critical transformation resistance: f, = 4.7 MJ/m.
The resulting nominal superelastic stress-strain curves using the material parameters listed above are also shown in Figures 2-20a, 2-20b and 2-20c. The quality of
the fit is very encouraging.
As mentioned previously, phase transformation during superelasticity typically
occurs by the nucleation and propagation of phase transformation fronts (Shaw and
"The permanent set observed in the experiment at 306 K is probably due to some plastic deformation at the higher stresses in this experiment.
50
Kyriakides, 1997). Figure 2-21 shows such a phenomenon in our numerical simulations. This figure shows the numerical stress-strain curve from the simulation for
300 K, along with the contours of the martensite volume fraction keyed to instances
(a) through
(j)
at different points along the superelastic stress-strain curve.
The
contours for (a) through (e) show that martensite nucleates at the right end of the
specimen where the gage width is the narrowest, and the transformation front propagates to the left until the gage section is fully transformed to martensite by stage (e).
The martensite volume fraction contours for (f) through
(j) show
that upon reversal
of imposed deformation the phase front recedes until the gage section is once again
fully austenitic by stage (j).
The results of experiments and corresponding finite-element predictions for superelastic experiments conducted at 300 K on sheet tensile specimens with axes oriented
along 450 and 900 to the rolling direction1 2 are shown in Figure 2-22a and 2-22b, respectively. The small amount of anisotropy in the superelastic stress-strain response
in these differently oriented specimens is well-captured by the predictions from the
constitutive model.
An important manifestation of the superelastic response of a shape-memory material is the strain-versus-temperature response at a fixed stress level. By cycling
the temperature over a narrow range at a fixed stress, one can obtain a recoverable
strain cycle. Figures 2-23a and b show experimental strain-temperature curves from
00-oriented specimens which were first subjected to fixed axial stress levels of 150 MPa
and 250 MPa, and then subjected to a temperature history where the temperature is
decreased at a constant rate of 0.016 K/s, and then increased back at the same rate.
With respect to Figure 2-23a, the initial elastic strain of ~ 0.005 is due to the axial
stress of 150 MPa. The material is initially in the austenitic state. As the temperature
is decreased from 315 K, the strain first decreases slightly due to thermal contraction,
and then at ~~276K the austenite-to martensite-transformation occurs and there is a
2
We have also investigated the behavior of Ti-Ni sheets which were thermo-mechanically processed
differently from the sheets studied in this section. In Appendix H we show that our constitutive
model is able to predict the anisotropy exhibited in the superelastic tension experiments on sheet
Ti-Ni conducted by Shan and Sung (2001) to good accord.
1
51
sudden burst of strain of the order 4%. Upon heating, there is at first a slight increase
in the strain due to thermal expansion, and then at ~ 302K the martensite transforms
back to austenite with a strain recovery of the order 4%. The temperatures at which
the transformations occur can be controlled by the initial stress bias. As shown in
Figure 2-23b, an increase in the initial stress increases the austenite-to-martensite
as well as the martensite-to-austenite transformation temperatures. Figure 2-23 also
shows that the corresponding predictions from the theory are in good agreement with
the experiments.
We next consider the coupled thermo-mechanical superelastic response of the
initially-textured Ti-Ni sheet. In order to examine the effects of heat generation
and conduction, we have performed superelastic tension experiments at three different nominal strain rates on 0W-oriented specimens at an initial temperature of 300 K.
In addition to the test at the "low" nominal strain rate of 3 x 10-5/sec reported previously, Figure 2-20b, where the response was expected to be (close to) isothermal, we
have conducted two additional experiments at slightly "higher" rates of 8.4 x 10
4
/sec
and 2 x 10-3/sec, where we expect to see some effects of the changes in temperature.
The superelastic stress-strain curves for all three strain rates are shown in Figure 2-24.
Note that as the strain rate increases, the stress-strain curves at the two higher strain
rates show a "hardening response". This apparent hardening due to thermal effects
is qualitatively similar to that reported previously by Entemeyer et al. (2000), who
have argued that the major contribution to the apparent hardening is due to temperature changes associated with the exothermic a -- m, and endothermic m -+ a
transformations.
Recall that the energy balance (2.21) requires computation of the term
div {J-1FK( )FTgrad
0}.
The finite-element program ABAQUS/Explicit (currently) limits user access to modify the computer code, and allows input of only a single scalar value for the thermal
conductivity. We have accordingly made several approximations in our calculations.
For the cubic austenite the thermal conductivity is isotropic, K' = 01. For the
52
monoclinic martensite the anisotropic thermal conductivities are not known, and as
with the elastic constants, these are also approximated as that for cubic materials,
K
m 1 . We shall use a constant value of
K =
(K" ±
m)/
2
in our numerical calcu-
lations. Further, since the superelastic strains are small, we neglect the contributions
from the J and FFT terms, and make the approximation div {J-1FK( )FTgrad 0}
r div grad 0. We also neglect the terms J-O Ee - C( )[A( )] + J-1 r in (2.21). Under
these approximative assumptions the equation governing the change in temperature
becomes
c
Kdiv grad0 +
Ai0
OT
+
fl.
(2.29)
Based on values published in the literature, we use the following values of
c = 2.1 MJ/m3 K,
s = 13W/m K
for Ti-Ni in our calculations. The calculations were carried out for a Ti-Ni sheet specimen modelled using the mesh shown in Figure 2-18b, this time using 446 C3D8RT
ABAQUS elements. Regarding the thermal boundary conditions, the whole mesh is
initially at 300K. The nodes at the grip ends are kept at 300K throughout the calculation to simulate massive hard grips which act as constant temperature baths. The
heat flux from the remaining faces of the tension strip due to convection in air is taken
to be governed by boundary conditions on the heat flux h in the deformed configuration, of the form h = h(6 - 0O)n, with a film coefficient h and a sink temperature
0. In our calculations we take the film coefficient to have a value h = 12 W/m 2 K,
and a sink temperature of 300K. Figure 2-24 also shows the resulting superelastic
stress-strain curves for all three strain rates. The hysteresis loop corresponding to
the lowest strain rate is very similar to that obtained from the isothermal calculation, Figure 2-20b, while the stress-strain curves calculated for the higher strain
rates show the experimentally-measured hardening response.
set
f,
-=
f, - --constant."
Recall that we have
Thus, the hardening response is entirely due to thermal
effects associated with the phase transformations. Figure 2-25 shows evolution of the
13
The interaction terms g3. have also been set to zero in our calculations.
53
contours of the martensite volume fraction at representative instances during the forward and reverse transformations for the test at the highest strain rate. Note that
because of the boundary conditions, both the forward and reverse transformations
start from the grip-ends and propagate as "fronts" towards the center of the specimen.
Figure 2-26 shows contours of the temperature at representative instances
during the forward and reverse transformations. During forward transformation the
temperature increases by as much as 22K from the ambient temperature of 300K due
to the exothermic austenite-to-martensite transformation, while it decreases by as
much as 22K from the ambient temperature during the reverse endothermic transformation from martensite-to-austenite.
The nucleation and propagation of phase
transformation fronts and the associated temperature changes in our calculations,
are qualitatively similar to the results reported by Shaw and Kyriakides (1997) for
their polycrystalline sheet tensile specimens.
The effects of heat conduction into and out off the gage section were further studied
by performing a superelastic tension-hold experiment. A representative strain-time
profile for such an experiment is as follows: the specimen is extended at a strain rate
of 1.25 x 10-3/sec with an intermediate hold of 10 sec, and then after the desired total
strain level the straining direction is reversed at a rate of -1.25 x 10-3 /sec, again with
an intermediate hold of 10 sec, Figure 2-27a. The experimental nominal stress-strain
curve and the corresponding finite-element prediction of this experiment are shown in
Figure 2-27b. The finite-element simulation nicely reproduces the stress-dip observed
in the experiment during the forward transformation, and the stress-increase during
the reverse transformation. The major cause for the stress-dip and stress-increase is
the heat conduction out of and into the specimen, respectively, during the forward
and reverse transformations.
54
3.00
2.62
2.23
1.85
XI
il
fill)
fill}
{110}
{110}
{110}
{100}
{100}
{100}
1.46
1.08
.69
.31
X2
(b)
(a)
(c)
PoPLa
Figure 2-1: (a) Experimentally-measured texture in the as-received Ti-Ni rod, (b)
its numerical representation using 729 unweighted discrete crystal orientations, and
(c) its numerical representation using 343 unweighted discrete crystal orientations.
Pole-figure data were obtained using PoPLa.
55
24
.I
I
I
I
,I
I
I
23
22
K
E
021
s-12.9 C
-)a
D
0
E 20
a)
v
C 19
w
018
-j
L-
-
t
Oaf= -4.70
60.2 C
-mf
-ms
21.9 C
!17
w
%
%
161151M
14L
-1 30
I
-140
I
-120
I
-100
I
I
I
-80
-60
-40
TEMPERATURE [C]
A I
I
-20
0
20
40
Figure 2-2: Differential scanning calorimetry thermogram for polycrystalline rod TiNi used in experiments.
56
0 =const.
Cr 4
a. I---------
I
poly
fc
ET
poy (
T(0 -0 T)
Ey9T
a
1
a
0'
6
Figure 2-3: Schematic of an isothermal superelastic stress-strain response in simple
tension.
57
0.475
-H
0.350>
+I
0.600
5.000
3/16R -+,k
Dimensions in inches
Figure 2-4: Specimen geometry for tension and compression experiments.
58
MART.
VOL.
FRAC.
0.48
0.54
-0.60
lii
l-0
.66
0 .71
0.77
'too,0
.83
0.89
0.94
1.00
(b)
(a)
1
I
HH_
I
900o EXPERIMENT
800-
-
FIT
700W~
0L
0
600-
W 500H0
400
300
200.
100
0
0
0.01
0.02
0.03
0 .04
STRAIN
I
0.05
I
0.06
0.07
(c)
Figure 2-5: (a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed
mesh at a tensile strain of 6%. Contours of martensite volume fraction are shown.
(c) Superelastic stress-strain curve in tension. The experimental data from this test
was used to estimate the constitutive parameters. The curve fit using the full finite
element model of the polycrystal is also shown.
59
MART.
FRAC.
VOL.
0 .00
0 .11
-
0.22
-0.55
0 .66
0 .77
o .89
1. 00
(b)
(a)
1000
900
o EXPERIMENT
-
FULL FEM
800
700
0
600
Co
500
I-
C',
400
300
200
100
0
)
0.01
0.02
0.04
0.03
STRAIN
0.05
0.06
0.07
(c)
Figure 2-6: (a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed
mesh at a compressive strain of 5%. Contours of martensite volume fraction are
shown. (c) Superelastic stress-strain curve in compression. The absolute values of
stress and strain are plotted. The prediction from the full finite-element model for
the polycrystal is also shown.
60
1000
900 800
0c.
-
TENSION
- -
COMPRESSION
700
600
CD)
W
w 500
U)
400
300
--
-
200
100
0
0.01
)
0.02
0.03
0.04
STRAIN
(a)
0.05
0.06
0.07
IVV
900
-
- -
800
TENSION
COMPRESSION
-/
700
CL)
II
-
LE
C/)
I
-
600
500
-/
-/
400
300
200 100
0
I
C
I
I
I
0.04
0.05
0.06
0.07
0.03
STRAIN
(b)
Figure 2-7: Comparison of the predicted response from tension and compression
simulations to demonstrate the numerically predicted tension-compression asymmetry
for (a) the polycrystal material and (b) single crystal oriented in the { 111 } direction.
0.01
0.02
61
3.00
2.59
2.19
1.78
1.38
.97
.57
.16
(a)
1000
900-
800-
-
-
TENSION
COMPRESSION
700$ 600CD,
CD 500C400
30020010000
0.01
0.02
0.04
0.03
STRAIN
0.05
0.06
0.07
(b)
Figure 2-8: (a) {111} pole figure corresponding to a random initial texture. (b)
Comparison of the predicted stress-strain response in tension and compression using
a random initial texture.
62
0.475
/1R
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Dimensons
ininche
0.475
Figure 2-9: Specimen geometry for torsion experiments.
63
- -
-
-
-
.--
____
-
~-
---------------
--
MART. VOL. FRAC.
-
0.17
- 0.26
- 0.35
-
0.44
0.54
0.63
0.72
-
0.81
- 0.91
- 1.00
(b)
(a)
500
450
o EXPERIMENT
400
-
FULL FEM
350
0l)
(0
0
300
w)
cc 250
FX0
W 200
150
100
50
0
Q01
0'
0.02
0.03
0.06
0.04 0.05
SHEAR STRAIN
0.07
0.08
0.09
0.1
(c)
Figure 2-10: (a) Undeformed mesh of 720 ABAQUS C3D8R elements. (b) Deformed
mesh at a shear strain of 9%. Contours of martensite volume fraction are shown. (c)
Superelastic stress-strain curve in torsion. The prediction from the full finite-element
model for the polycrystal is also shown.
64
n n.
Axial strain
Shear strain
-
0.051
.,
0.04
-
.'-
"0.03
\-
0.02
0.01
MART .
0 00
20
300 400
500 600
TIME [s]
700
800
FRAC .
VOL
900 1000
(a)
0.56
0 .67
0.89
11. 00
-0.78
( b)
ROn
o
5
EXPERIMENT
PREDICTION
-
(L 4(
0-
0000
(c)
350
300
A
c
EXPERIMENT
- - PREDICTION
U,
_0%onn000000
AA
250
200
AAA
A
50)
W 3(
CC
F-
0
U)
0
0
000
.0
w 150
00
A'
-4
0
2( 0 .
0
0.005
.o
.0
0.01
0.015
A-
A A
AA
50
0-
A
100
1U
.
AA
A - A'
A
.0
A
A
0.02 0.025
STRAIN
0.03
0.035 0.04
o 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
SHEAR STRAIN
(d)
(e)
Figure 2-11: (a) Loading program for combined tension-torsion experiment. (b) Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed mesh at a tensile
strain of 3% and a shear strain of 4.5%. Contours of martensite volume fraction are
shown. (d) Superelastic stress-strain curve in tension. (e) Superelastic stress-strain
curve in shear. The prediction from the full finite element model for the polycrystal
is also shown.
65
0.1
0.09,
-
0.08-
Axial strain
Shear strain
0.07.
0.06.
0.050.04
I
0.03.
.I
00
100 40
00 50
60
70
80
0.02.
0.01
0
MART . VOL . FRAC .
0
100 200
300 400
500 600 700
TIME [s]
800
900 1000
0.00
0 .11
0.22
.0.33
(a)
0.67
0.78
0.89
11.00
(c)
(b)
700m.
.
* EXPERIMENT
-- o- PREDICTION
6001
a-
500
.
.
A
400
0
500
.0
w
0
H300
-000*
200|
0
0
0
C 200
.
-
w
W
S 000
100
.
0
.0.0
-00 0.0
.000
0.0
0
0
-
EXPERIMENT
PREDICTION
0
.
100
2 300
w
c400
-n
--
0.005 0.01
0
00*
0.015 0.02 0.025
0.03
.
0.035 0 .04
-1001-0
STRAIN
.01
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.08
SHEAR STRAIN
(e)
(d)
Figure 2-12: (a) Loading program for the path change tension-torsion experiment. (b)
Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed mesh at a tensile
strain of 3% and a shear strain of 7%. Contours of martensite volume fraction are
shown. (d) Superelastic stress-strain curve in tension. (e) Superelastic stress-strain
curve in shear. The prediction from the full finite element model for the polycrystal
is also shown.
66
1000
900
0 800
co
C)
w
CE
ICl)
o EXPERIMENT
FULL FEM
700
600
500
400
300
200
100
0
--
---------------------
)
N
STRAIN
(a)
900
2
I
0.01 0.02 0.03 0.04 0.05 0.06 0.07
1000
CL
- TAYLOR MODEL
800
700
.
o EXPERIMENT
---
FULL FEM
TAYLOR MODEL
600
w 500
-------- ---
C,)
Cl
400
300
200
100
0
It
C/)
Cn
Cf)
w
CC
Cl)
Ic
w
M
Cl)
500
450
400
350
300
250
200
150
100
50
0
.------- -----
0.01 0.02 0.03 0.04 0.05 0.06 0.07
STRAIN
(b)
. . . . . . . . .
o EXPERIMENT
FULL FEM
---
TAYLOR MODEL
..--
0.02
--
- - -- -
0.04 0.06 0.08
SHEAR STRAIN
(c)
0.1
Figure 2-13: Comparison of the Taylor model against experiment and full finiteelement calculation in (a) simple tension, (b) simple compression, and (c) shear.
67
700.
1
1
1
-
600
I
o EXPERIMENT
- TAYLOR MODEL
+ FULL FEM
500
--
400
U)
w
a:_ 300
-
Hn
0
0o
-
200
000
00
100
0
0
0.005
0.01
0.015
0.02
STRAIN
0.025
0.03
0.035
0.04
(a)
500
A
400
-+
a-L
-
EXPERIMENT
TAYLOR MODEL
FULL FEM
300U)
U)
200U)
100
0
-io IiI
-
0.01
0
0.01
0.02
0.03
0.04
0.05
SHEAR STRAIN
(b)
0.06
0.07
0.08
Figure 2-14: Response of the Taylor Model compared against the path change tensiontorsion experiment and full finite element calculation in (a) tension and (b) shear.
68
800
0z
o EXPERIMENT
700-
FULL FEM
- - - TAYLOR MODEL
-
600
00
W400.
0
o:o*
300
200
oooa
****--*
**~~
100
0
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
STRAIN
(a)
800
700-
'ts
- _,----- .----------------------
S600-
..-
U) 500
00;C
'
0
W
**'
c400
0
I-
Cn 300
,.....--0o-
200
o EXPERIMENT
FEM
-FULL
--- TAYLOR MODEL
100
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
STRAIN
(b)
800'
---------------
700
''
FULL FEM
-
2 600-
,
011
*,,'
(500
400- *-.-C/
300
0/
200
100
C
0*
0.0
00
.
2
00
o EXPERIMENT
-- FULL FEM
--- TAYLOR MODEL.
.
4
00
.
6
00
0.01 0.02 0.03 0.'04 0.05 0.06 0. 07
STRAIN
(c)
Figure 2-15: Superelastic stress-strain curves in tension (Shaw and Kyriakides, 1995)
at three temperatures (a) 6 = 333.2 K, (b) 0 = 343.2 K, and (c) 0 = 353.2 K.
Full finite-element and Taylor model prediction from the constitutive model are also
shown. The material parameters for the Ti-Ni of Shaw and Kyriakides is obtained
from the data at 343.2 K.
69
OB
o EXPERIMENT
-SIMULATION
3000
250N
=00
200,
Cl)
150
100
w
00
0 < A
W-
0*0. .326'
00
350,
030.2
3049
MPRAUR
[K]
3106
TE
STRAIN
Figure 2-16: Simulation of austenite-martensite-austenite shape memory effect (Shaw
and Kyriakides, 1995): Isothermal stress-strain response for straining at constant
strain rate (ABC) at 0 = 303.2 K followed by a temperature (CD) to 0 = 305.8 K.
70
- -
3.00
2.60
2.20
1.80
1.41
1.01
.61
.21
I
-----.------
~
---
-
-~
TD
fill})
{111})
{110}
{110}1
{100}
{1Ut
(a)
(b)
Figure 2-17: (a) Experimentally-measured texture in the as-received Ti-Ni sheet; and
(b) its numerical representation using 48 weighted crystal orientations.
71
20
4.2
4
5.72
)P
3.65
3.85
I
A I
0.53
S5.72
31
All dimensions in mm
(a)
(b)
Figure 2-18: (a) Geometry of the tensile specimen (drawing is not to scale). (b)
Undeformed finite-element mesh of the tensile specimen using 446 ABAQUS threedimensional elements. The gage width of the specimen is linearly tapered from
3.85 mm to 3.65 mm as shown in (a).
72
Ti
T (0 -
am
T'(01)
am
-
0
)
T
-
T
,/
0
/
T
ma
03)
fc
'
(02)
(O')ma
,'
0
ms
-'
T
ma (01)
0
0
as
Figure 2-19: Schematic diagram of the variation of resolved forces on an active transformation system i with temperature.
73
450
400
o EXPERIMENT:
350
--
; 289 K
FIT
300
c 250
CO
w
a: 200
-0,
0~
01
00
150
100
bb81
J.
MOM
50
0
0
0.01
0.02
0
RAR0
04
0.05
0.06
(a)
4501
400
350
001
300
0 1
U,
U,
I00
250
200
---- F0T
F-
U,
150
100
50
0
0
40
0.01
0.02
0
0.03
0.04
STRAIN
0.05
0.06
(b)
450
*
--
400
t
350
/
/0
EXPERIMENT: 00300K
/
300
U,
250
S--
200
FIT
F-
U)
150
100
50
0
o
of
0
0
0
0.01
0.02
0.03
0.04
STRAIN
0.05
0.06
(c)
Figure 2-20: Nominal superelastic stress-strain curves from tension tests on 0'oriented specimens. The tests were conducted under isothermal conditions at (a)
289 K, (b) 300 K, and (c) 306 K. The experimental data from these tests were used to
estimate the phase transformation parameters. The curve fit from the finite-element
simulations is also shown.
74
-
- __ I
' .4 m_ - - -
' _= - &._111
__
-
__
- __
- - -
__
500
4501400
--
350
CO,
--
d
-
d/
.
f
300
Cl,
w)
W
-
250
II
-
200
h
150
I
-- --
-I
100
50
0j
0.01
0.02
0.03
0.04
STRAIN
(a)
(b)
0.05
0.06
0.07
0.08
(f)
1.00
0.89
0.78
(g)
0.67
0.56
0.44
0.33
(C)
0.22
0.11
(h)
0.00
(d)
(i)
(e)
(j)
Figure 2-21: A superelastic finite-element simulation of an experiment conducted
under isothermal conditions at 300 K. Contour plots of the martensite volume fraction are also shown. The set of contours on the left shows the transformation from
austenite-to-martensite during forward transformation, while those on the right show
the transformation from martensite-to-austenite during reverse transformation. Note
that because of the slight taper in the gage width, the austenite-to-martensite transformation nucleates at the right grip-end and propagates toward the opposing gripend.
75
---
__
_
-
. 41W
450
40035000/
/
25
I
-
200 -
150
-
-
U)
100 0
50 -
-
0
0.01
0.02
-
EXPERIMENT : 45 ; 300 K
PREDICTION
0.03
0.04
0.06
0.05
STRAIN
(a)
450
400-1
-
350 --
Co50
300
E
4
0
o1/
0_
250 200 -
-
0
300C
/0O
100 -
'
o
0 EXPERIMENT :90 ;300 K
50
-
PREDICTION
50 -
0
-
0.01
0.02
0.03
0.04
0.05
0.06
STRAIN
(b)
Figure 2-22: Experimental superelastic stress-strain curves in tension at 300 K: (a)
450 orientation; (b) 90 orientation. The corresponding predictions from the finiteelerment simulations are also shown.
76
0.09
0.080
-
0.07-
-
EXPERIMENT : 0; 150 MPa
PREDICTION
0.06-
z
Cl)
0.050-
0.04-
0
0.03-
10
0
0~
0 1
o
I
01
01
0.02
I
t
~-pJ0
0.01 I
0
270
280
300
310
TEMPERATURE [K]
290
320
330
340
(a)
0.09
0.08
-
o
0.07
-
0.06
-
-
-
EXPERIMENT : 09 250 MPa
PREDICTION
~z
0.05 I
C/)
0.04-
01
I-
01
10
1
0.03-
10
1o
01
0'1
0.02-
010
0
10
0.01
0
340
320
330
310
300
TEMPERATURE [K]
(b)
Figure 2-23: Strain-temperature cycling experiments conducted on 0-oriented speci270
280
290
mens at pre-stress levels of (a) 150 MPa; and (b) 250 MPa. The predictions from the
finite-element simulations are also shown.
77
600
.
.
I
I
.
o EXPERIMENT: 0.00003 /s
A EXPERIMENT: 0.00084 /s
o EXPERIMENT: 0.002 /s
500 I
400 1
a
[313
-3E:000
.03/s
Ca
Cl)
C/)
300 F
-3
F E1:
C/)
200 1
-
-
100
~~ ~
~
-~~ ~
--.-- ~
------ a0
0
0.01
0.02
~
0.03
~ E 1:*003/
~ -EM0008/
--
0
-
C
~
0.05
0.04
STRAIN
0.
-
0.06
FE
:
.000 4 /
FEM: 0.0020/s
0.07
0.08
0.09
Figure 2-24: Superelastic stress-strain curves in tension along the rolling direction at
nominal strain rates of 2 x 10-3 /sec, 8.4 x 10-4 /sec, and 3 x 10' /sec ("isothermal
condition"). The predictions from the theory are also shown.
78
700
6001-
e
500
if
-
C
Cl)
400
Cl)
b
300
-
-
200
100
--
0
j
0.01
0.02
0.03
0.05
0.04
STRAIN
(a)
(b)
0.06
0.07
0.08
0.09
(f)
1.00
0.89
0.78
(g)
0.67
(C)
0.56
0.44
0.33
0.22
0.11
0.00
(h)
(d)
(i)
(e)
(j)
Figure 2-25: A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10-3 /sec. The set of the martensite volume fraction contours on the left show the transformation from austenite-to-martensite during forward
transformation, while those on the right show the transformation from martensite-toaustenite during reverse transformation. Note that because of the boundary conditions, both the forward and reverse transformations nucleate from the grip-ends and
propagate towards the center of the specimen.
79
70 0
60 0-
e
50 0C,,
2l
w)
0---
40
(-
b
Wl
30 0dh
20 0
--
o
II
'g1
10
0
j 0.01
0.02
0.03
0.05
0.04
STRAIN
(a)
0.06
0.07
0.08
0.09
(f)
0
(b)
(C)
322 .0
317.1
312.2
307.3
302.4
297.6
292.7
287.8
282.9
278.0
(g)
(h)
(d)
(i)
(e)
(J)
Figure 2-26: A superelastic finite-element simulation of an experiment conducted
at a nominal strain rate of 2 x 10-3 /sec. The set of contours on the left shows the
temperature increase during transformation from austenite-to-martensite, while those
on the right show the temperature decrease during transformation from martensiteto-austenite. During forward transformation the temperature increases by as much
as 22'K above the ambient temperature of 300 K due to the exothermic austeniteto-martensite transformation, whereas it decreases by as much as 22'K below the
ambient temperature during the reverse endothermic transformation from martensiteto-austenite.
80
0.07
0.06
0.05
z 0.04
Cc
0.03
0.02
0.01
0
20
0
40
60
100
80
TIME [s]
(a)
120
140
700
600 1-
o EXPERIMENT: 0
- - PREDICTION
500
co
400
C/)
__00 - - o
oooooooooog.
-
(I)
300
000000000
0
-0
0
P
200
100
-00".o
01 -1
01 0
C
o
o
0o00000000000
/i
~
- -0
0,0
----- L
0.01
,..--000
000oo
0.02
0
0
/
,0
0~
,
0
o0oo
0.03
0.04
STRAIN
(b)
0.05
0.06
0.07
Figure 2-27: (a) Nominal strain versus time profile for the tension-hold experiment
at a baseline nominal strain rate of 1.25 x 10' /sec and initial temperature of 300 K.
(b) Superelastic stress-strain curve measured in the experiment. The prediction from
the finite-element simulation is also shown.
81
Chapter 3
Isotropic-based constitutive model
3.1
Constitutive model
In this section we summarize our three-dimensional constitutive model for the
isotropic superelastic response of shape-memory materials. This model is meant to
characterize small elastic strains in the presence of temperature fields close to a fixed
reference temperature 60. The constitutive framework is similar in spirit to the crystalmechanics based theory of Anand and Gurtin (2002).
Using the standard notation of modern continuum mechanics, 1 we recall that the
deformation gradient F maps referential segments dX to segments dx = FdX in the
deformed configuration. We base the theory on the following multiplicative decomposition of the deformation gradient: F = FeFP. Here (i) FP(X) represents the local deformation of referentialsegments dX to segments dl = FP(X)dX in the relaxed lattice
configurationdue to the generation, growth and annihilation of austenitic/martensitic
fine structure in a microscopic neighborhood of X. (ii) F (X) represents the mapping
of segments dl in the relaxed lattice configuration into segments dx = FI(X)dl in
'Notation: V and Div denote the gradient and divergence with respect to the material point
X in the reference configuration; grad and div denote these operators with respect to the point
x = y(X, t) in the deformed configuration, where y(X, t) is the motion; a superposed dot denotes
FP-T = (FP) T, etc.. For any
the material time-derivative. Throughout, we write Fe 1 = (Fe)-',
tensor A, symA = (1/2)(A + A T ), skwA = (1/2)(A - A T), and A 0 , the deviatoric part of A, is
defined by Ao = A - (1/3)(tr A)1; A is deviatoric if tr A = 0. Also, the inner product of tensors
A and B is denoted by A -B, and the magnitude of A by JAl = v'A- A.
82
the deformed configuration due the "elastic mechanisms" of stretching and rotation
of the lattice.
We denote by
< 1, the local volume fraction of martensite at X.
(X, t), 0 <
The evolution of FP is taken to be governed by the transformation rate
relation FP = DPFP, DP =
via the
NP. where NP is a symmetric transformation tensor
3/2ET, where ET is the transformation strain in simple
with magnitude INPI =
tension. Transformationfrom austenite to martensite, abbreviated a -+ m, occurs
when c > 0; and transformation from martensite to austenite, abbreviated m -* a,
occurs when
< 0. Shape-memory materials show a slight inelastic volume change
during transformation, and this is well-described by the crystal-mechanics theory.
However, for the phenomenological isotropic theory under consideration, we shall
neglect any small inelastic volume changes and assume that det FP = 1; accordingly
tr NP = 0.
The underlying constitutive equations relate the following basic fields:
free energy density per unit referential volume,
F,
det F > 0,
deformation gradient,
0,
0 > 0,
absolute temperature,
T,
T = T T,
Cauchy stress,
q,
heat flux per unit referential area,
FP,
Fe = FFP-,
det FP = 1
transformational part of the deformation gradient,
0 <
martensitic volume fraction,
K
det Fe > 0,
Ce = FeTFe,
E
=
(1/2) (Ce
T
=
ReTTRe,
1,
elastic part of the deformation gradient,
elastic right Cauchy-Green strain,
-
1),
elastic strain,
stress conjugate to E .
The constitutive equations are:
83
1. Free energy: The free energy is taken in the separable form
O(Ee,O, ) = Oe(Eeo, )+
0(O) +,Otr(o,
e(Ee, 0,() = (1/2)Ee - C()[Ee](
-
),
with
00 )A( ) - C( )[Ee],
V)O(0) = c(0 - Oo) - cO In (0/0o) ,
V) tr(0, ) = (AT/0r) (0
-
OT)
(3.1)
(3.2)
(3.3)
(3.4)
(.
Here
C(O) =
E(()
E()
+
(I + V(0))I
101,(
v(()
10 1
O
(1 - 2V( ))_'
(3.5)
with E and v the Young's modulus and Poisson's ratio is the elasticity tensor,
and
A( )
=
a(c) 1,
(3.6)
with a the thermal expansion coefficient is the thermal expansion tensor, at the
reference temperature 60. The constant c is the specific heat per unit volume
in the reference configuration. The parameter
perature,
AT
0
T
is the phase-equilibrium tem-
is the 'latent heat of the a +-+ m transformation at temperature
OT.
2. Stress: The elastic stress-strainrelation has the form
Te -
-~C( )[E' - A( )(O - 00)].
(3.7)
3. Resolved force. Back force. Dissipative force: The quantity
5= Te - NP
(3.8)
is a resolved force, with NP the transformation tensor (defined below).
The
derivative
b=
= (AT/OT)(O -
OT)
(3.9)
defines an energetic backforce. Let
f= (-
84
b),
(3.10)
then thermodynamic considerations show that
f ;> 0;)(3.11)
accordingly
f is called the
dissipative force2 associated with the phase transfor-
mation.
4. Transformation conditions: The transformation conditions are
f,
for
> 0,
(3.12)
f = -f,
for
< 0,
(3.13)
f=
with
ft
> 0 a constant transformation resistance.
5. Flow rule: The flow rule is defined by an evolution equation for FP of the form
FP = DPFP,
JNP,
DP =
with martensitic volume fractions
(3.14)
consistent with 0 <
< 1, and where the
transformation tensor NP is given by
Te
V(3/2) ET
if
=0,
NP =(3.15)
(3/2) ET
Te(tc)
if 0 <
0
ITe (tc) I
<
1.
Here ET is a constant transformationstrain in simple tension. Also, beginning in
an austenitic state
volume fraction
(
= 0), the time t, denotes the instant when the martensite
becomes positive. 3 Moreover,
2
Also
3
called the driving force for phase transformation.
1n Appendix G we show that if following classical isotropic plasticity we take
NP = f(3/2)Te
ITe I
for all
O 5(
1,
then we get spurious results in superelastic deformations in simple shear. Whereas, by fixing the
transformation direction during a superelastic forward and reverse event to be determined by the
stress at the first instance of transformation, yields acceptable results, at least for the proportional
loading conditions tested in this section.
85
(i) if -fc < f < fe, then
(ii) if
f = f, and
0 <
'
= 0;
< 1, then
and
(iii) if
f
= -f,
and 0 <
(3.16)
(f + fc) ; 0;
(3.17)
< 1, then
and
(iv) if f=fc and
(f - fc) < 0;
=1, or if f = -f, and
=
0, then
=
0.
The consistency conditions (3.16) and (3.17) serve to determine the transformation rates
.
6. Entropy; energy balance; heat flux:
The entropy rq is given by
-
09
)
-=
cln (9/9) + Ee -C( )[A(()J
-
(AT/OT)
,
(3.18)
the balance of energy may be written as
0
(3.19)
= -Divq + f + r,
where r is the heat supply per unit volume of the reference configuration, and
the referential heat flux q is taken to be governed by Fourier's law
q = -K( )VO,
where K() is the thermal conductivity tensor at
(3.20)
00.
Using (3.18) and (3.20),
the energy balance (3.19) becomes
cO = Div {K( )VO} +
((AT/OT)
0 + f)
- 0 Ee -C( )[A( )] + r.
(3.21)
To complete the theory for a particular material the constitutive parameter/functions that need be specified are
{ E (),
v ( ), a ( ), CT,
86
OT, AT,
fc, C,K(D
Finally, the standard partial differential equation for the stress referred to the
deformed configuration is
div T + J-1 f
=
(3.22)
0,
where div represents the divergence operator in the deformed configuration, J = det F
and f is the body force per unit volume in the reference configuration (the body force
f is assumed to include inertial forces). The differential equation for the temperature
referred to the deformed configuration is
J- 1 cO = div J-'FK( )Fgrad 0} + J-1
((AT/OT) 0
+ f)
- J-1 OEe -C( )[A( )] + J- 1 r.
(3.23)
We have implemented our constitutive model in the finite-element computer programs ABAQUS/Explicit and ABAQUS/Standard (2001)' by writing user material
subroutines. Algorithmic details of the numerical implementation are given in Appendix E. 5
4 All the calculations performed in this work were done using ABAQUS/Explicit (2001) unless
stated otherwise. The results using both schemes match identically e.g. the finite-element simulations using ABAQUS/Explicit and ABAQUS/Standard for the coupled-temperature displacement
analysis for the polycrystalline sheet material in Section 3.2.1 is shown in Figure 3-1.
'All calculations reported in this section were performed using ABAQUS/Explicit, while those
reported in Section 3.3 were performed using ABAQUS/Standard.
87
3.2
Evaluation of the constitutive model for polycrystalline Ti-Ni alloy
3.2.1
Sheet material
Suitably processed 0.53 mm thick Ti-Ni sheets, intended for superelastic behavior at
room temperature, were obtained from a commercial source. Recall that the material
parameters that need to be specified for the theory are
1. The thermo-elastic parameters {E( ), v(s), a( )}.
2. The phase transformation parameters {ET,
OT, AT,
fc}.
3. The specific heat c and the thermal conductivity s().
The values of {E, v, a} for the austenitic and martensitic phases of Ti-Ni are welldocumented in the literature. We have used the following values in our calculations:
Young's modulus and Poisson's ratio for austenite: Ea = 62 GPa, Va = 0.33,
Young's modulus and Poisson's ratio for martensite: Em = 31 GPa, Vm = 0.33,
Coefficients of thermal expansion: aa
=
11 X 10- 6 /K,
0m = 6.6 x 10- 6 /K,
and assumed a simple rule-of-mixtures dependence for the dependence of E( ) and
a( ) on the volume fraction of martensite
.
An estimate for the phase transformation parameters {ET,
6
T,
AT,
fc}
for Ti-Ni is
obtained by fitting the constitutive model to a set of superelastic tension experiments
at (at least) three different temperatures spanning the range of temperatures of interest. With reference to Figure 3-2, an operational procedure to estimate the material
parameters is as follows:
First, estimate the transformation strain eT from the superelastic tension experiments at each temperature. The measured transformation strains ET will typically
be a a function of temperature. However, for applications, a constant value of ET
88
which represents an average over the range of temperatures is usually an adequate
approximation.
Next, estimate the stress a-am for forward transformation, and the stress oma for
reverse transformation at each temperature. A schematic plot of the variation of -am
and Uma with temperature 9 is shown in Figure 3-2b. From this, the intercept of the
line aam versus 9 at zero stress defines the martensite start temperature 0ms, while the
intercept of the line -ma versus 9 at zero stress defines the austenite start temperature
0as; the mean value
1
OT=
2
(3.24)
(as + Oms)
determines the phase equilibrium temperature. From (3.9), (3.10), (3.12) and (3.13)
we note that during forward transformation
-
am(()
( -
(3.25)
OT
CT
(CT OT
holds, while during reverse transformation
Uma(0) -
(
T
(ET OT)
holds. Thus, the transformation resistance
(9
fc
-
OT) =
(3.26)
ET
at each temperature is given by
1
f,
=
(3.27)
- (Cam(9) - 0ma(9)).
CT X
2
Further, the mean values of -am and
na
..
at each temperature vary linearly with
temperature, i.e.
I(-am()
2
Thus, the slope of the line
(A
+ Uma(9))
1(0-am(9)
(
-
(3.28)
T).
ET OT
+ oma()) versus 9 gives
2
T
which in turn
\ET6T /
gives the latent heat of phase transformation, AT.
To evaluate these phase transformation parameters for the Ti-Ni sheet, superelastic tension experiments were conducted on polycrystalline sheet specimens with the
tension axis aligned along the sheet rolling direction, 0W-orientation. The geometry of
the tension specimens is shown in Figure 2-18a. The experiments were conducted at
three different temperatures, 289 K, 300 K and 306 K, under displacement control at a
89
very low nominal strain rate of 3 x 10
5
/sec, to ensure (near) isothermal conditions.
The nominal stress-strain curves from these experiments are shown in Figures 3-3a,b,
and c, respectively 6 . Following the procedure outlined above, the value of the estimated phase transformation parameters is
Phase equilibrium temperature: OT
Latent heat:
AT =
=
271 K;
110 Mj/m 3 ;
Critical transformation resistance: f, = 4.15 MJ/m 3 ;
Transformation strain:
ET =
0.047.
Corresponding numerical simulations of these superelastic experiments, using the
material parameters listed above, were carried out on the finite-element mesh (446
ABAQUS-C3D8R elements) shown in Figure 2-18b. The numerically-calculated nominal stress-strain curves are also shown in Figure 3-3. Although, as anticipated, a
constant value of ET over the three temperatures results in some discrepancy, the
overall quality of the fit using a simple isotropic model is very encouraging.7
As mentioned previously, phase transformation during superelasticity typically
occurs by the nucleation and propagation of phase transformation fronts. Figure 34 shows such a phenomenon in our numerical simulations. 8 This figure shows the
numerical stress-strain curve from the simulation for 300 K, along with the contours
of the martensite volume fraction keyed to instances (a) through (j) at different points
along the superelastic stress-strain curve. The contours for (a) through (e) show that
martensite nucleates at the right end of the specimen where the gage width is the
narrowest, and the transformation front propagates to the left until the gage section is
fully transformed to martensite by stage (e). The martensite volume fraction contours
for (f) through (j) show that upon reversal of imposed deformation the phase front
recedes until the gage section is once again fully austenitic by stage (j).
'We use nominal values of stress and strain since the phase transformation occurs in an inhomogeneous fashion along the gage section of the specimen. The permanent set observed in the experiment
at 306K is probably due to some plastic deformation at the higher stresses in this experiment.
7The crystal mechanics model used in Chapter
2 gives a much better agreement with experiments,
but at a higher computational expense.
8
A geometric "imperfection" in the form of a slight taper along the gage section is given to
provide an initiation site for phase transformation.
90
An important manifestation of the superelastic response of a shape-memory material is the strain-versus-temperature response at a fixed stress level. By cycling
the temperature over a narrow range at a fixed stress, one can obtain a recoverable
strain cycle. Figures 3-5a and b show experimental strain-temperature curves from
0 0-oriented specimens which were first subjected to fixed axial stress levels of 150 MPa
and 250 MPa, and then subjected to a temperature history where the temperature is
decreased at a constant rate of 0.016 K/s, and then increased back at the same rate.
With respect to Figure 3-5a, the initial elastic strain of ~ 0.005 is due to the axial
stress of 150 MPa. The material is initially in the austenitic state. As the temperature
is decreased from 315 K, the strain first decreases slightly due to thermal contraction,
and then at ~~279K the austenite-to martensite-transformation occurs and there is a
sudden burst of strain of 4.7%. Upon heating, there is at first a slight increase in the
strain due to thermal expansion, and then at ~ 300K the martensite transforms back
to austenite with a strain recovery of 4.7%. The temperatures at which the transformations occur can be controlled by the initial stress bias. As shown in Figure 3-5b,
an increase in the initial stress increases the austenite-to-martensite as well as the
martensite-to-austenite transformation temperatures.
We next consider the coupled thermo-mechanical superelastic response of the TiNi sheet. In order to examine the effects of heat generation and conduction, we have
performed superelastic tension experiments at three different nominal strain rates
on 0-oriented specimens at an initial temperature of 300 K. In addition to the test
at the "low" nominal strain rate of 3 x 10- 5 /sec reported previously, Figure 3-3b,
where the response was expected to be (close to) isothermal, we have conducted two
additional experiments at slightly "higher" rates of 8.4 x 10-4/sec and 2 x 10 3 /sec,
where we expect to see some effects of the changes in temperature. The superelastic
stress-strain curves for all three strain rates are shown in Figure 3-6. Note that as
the strain rate increases, the stress-strain curves at the two higher strain rates show a
"hardening response". This apparent hardening due to thermal effects is qualitatively
similar to that reported previously by Entemeyer et al. (2000), who have argued that
the major contribution to the apparent hardening is due to temperature changes
91
associated with the exothermic a ->
m, and endothermic m --+ a transformations.
Recall that the energy balance (3.23) requires computation of the term
div J-1FK( )FTgrad0}.
The finite-element program ABAQUS/Explicit and ABAQUS/Standard (currently)
limits user access to modify the computer code, and allows input of only a single
scalar value for the thermal conductivity. We have accordingly made several approximations in our calculations. We shall use a constant value of thermal conductivity
k = (k' + k')/2 in our numerical calculations, where ka and km is the polycrystalline
thermal conductivity coefficient for the austenite and martensite phase, respectively.
Further, since the superelastic strains are small, we neglect the contributions from
the J and FFT terms, and make the approximation div{ J
1
FK( )FT grad }
1 r in (3.23). Under
k div grad 0. We also neglect the terms J-lOEe -C( )[A(()] + J--
these approximative assumptions the equation governing the change in temperature
becomes
c
k div gradO +
(
+f
(OT
)
.(329)
(.9
Based on values published in the literature, we use the following values of
c = 2.1 MJ/m3 K,
k = 13W/m K
for Ti-Ni in our calculations. The calculations were carried out for a Ti-Ni sheet specimen modelled using the mesh shown in Figure 2-18b, this time using 446 C3D8RT
ABAQUS elements. Regarding the thermal boundary conditions, the whole mesh is
initially at 300K. The nodes at the grip-ends are kept at 300K throughout the calculation to simulate massive hard grips which act as constant temperature baths. The
heat flux from the remaining faces of the tension strip due to convection in air is taken
to be governed by boundary conditions on the heat flux h in the deformed configuration, of the form h = h(9 - OO)n, with a film coefficient h and a sink temperature 0 0.
In our calculations we take the film coefficient to have a value h = 12 W/m 2 K, and
a sink temperature of 300K. Figure 3-6 also shows the resulting superelastic stressstrain curves for the two higher strain rates. The stress-strain curves calculated for
92
the higher strain rates show the experimentally-measured hardening response. Recall
that we have set
fc
to be constant. Thus, the hardening response is entirely due to
thermal effects associated with the phase transformations. Figure 3-7 shows evolution
of the contours of the martensite volume fraction at representative instances during
the forward and reverse transformations for the test at the highest strain rate. Note
that because of the boundary conditions, both the forward and reverse transformations start from the grip-ends and propagate as "fronts" towards the center of the
specimen. Figure 3-8 shows contours of the temperature at representative instances
during the forward and reverse transformations. During forward transformation the
temperature increases by as much as 25K from the ambient temperature of 300K due
to the exothermic austenite-to-martensite transformation, while it decreases by as
much as 25K from the ambient temperature during the reverse endothermic transformation from martensite-to-austenite.
The nucleation and propagation of phase
transformation fronts and the associated temperature changes in our calculations,
are qualitatively similar to the results reported by Shaw and Kyriakides (1997) for
their polycrystalline sheet tefisile specimens.
The effects of heat conduction into and out of the gage section were further studied
by performing a superelastic tension-hold experiment. A representative strain-time
profile for such an experiment is as follows: the specimen is extended at a strain rate
of 1.25 x 10-3/sec with an intermediate hold of 10 sec, and then after the desired total
strain level the straining direction is reversed at a rate of -1.25 x 10-3 /sec, again with
an intermediate hold of 10 sec, Figure 3-9a. The experimental nominal stress-strain
curve and the corresponding finite-element prediction of this experiment are shown in
Figure 3-9b. The finite-element simulation nicely reproduces the stress-dip observed
in the experiment during the forward transformation, and the stress-increase during
the reverse transformation. The major cause for the stress-dip and stress-increase is
the heat conduction out of and into the specimen, respectively, during the forward
and reverse transformations.
93
3.2.2
Rod material
Suitably thermo-mechanically processed and heat-treated Ti-Ni at 55.96 wt.% Ti
drawn rods of 12.70 mm diameter, intended for superelastic applications, were obtained from a commercial source.
The thermo-elastic constants for the rod-material is chosen to be the same as in
Section 2.3.1. Using the measured phase transformation temperatures (Figure 2-2),
the value for the phase equilibrium temperature, OT, is given by
T
=(1/2) {Oms + 0,} = 256 K.
To obtain the remaining values of the phase transformation constants {AT,
6
T,
fc},
the constitutive model is fitted to a superelastic experiment in simple tension at
0 = 298 K, Figure 3-10a. The experiment is conducted at a very low strain rate to
ensure near-isothermal testing conditions.
The finite-element simulation in simple tension is performed using a single
ABAQUS-C3D8R element.
Using the procedure for material parameter determi-
nation outlined above the fitted phase transformation parameters are
Latent heat: AT
=
100 MJ/m3 ;
Critical transformationresistance: f, = 7.6 MJ/m 3;
Transformation strain: CT
=
0.05.
The resulting superelastic stress-strain curve using the material parameters listed
above are also shown in Figure 3-10a. The quality of the fit is very encouraging.
With the material parameters calibrated we proceed to perform a tubular-torsion
experiment on a thin-walled specimen performed at a very low shearing strain rate.
The stress-strain response in torsion at 0 = 298 K is shown in Figure 3-10b. The
finite-element simulation in simple shear is conducted using a single ABAQUS-C3D8R
element. The predicted nominal shear stress-strain curve is shown in Figure 3-10b,
where it is compared against the corresponding experimentally-measured curve from
a tubular torsion experiment. The prediction is in good accord with the experiment.
Proportional-loading tension-torsion experiment:
94
The strain-history imposed on a thin-walled tubular specimen during the proportionalloading tension-torsion experiment is shown in Figure 3-1 la. The axial strain c is
increased at a constant rate from 0% to 3% in 500 seconds, while the shear strain -y is
increased at a constant rate from 0% to 4.5% in the same time period. These strains
are then linearly reduced to zero, in another 500 seconds.
The initial finite-element mesh using 768 ABAQUS-C3D8R elements for the combined tension-torsion simulation is shown in Figure 3-11b, and the finite-element mesh
after 3% strain in tension and 4.5% strain in shear is shown in Figure 3-11c, along
with the contours of the martensite volume fraction.
Predictions of the axial-stress versus axial-strain, and shear-stress versus shearstrain from the constitutive model are compared against corresponding experimental measurements conducted at 0 = 298K in Figures 3-11d and 3-11e, respectively.
The measured superelastic stress-strain response in this tension-torsion experiment is
moderately well-approximated by the predictions from the constitutive model.
95
3.3
Deformation of a Ti-Ni biomedical stent
The superelastic response of Ti-Ni, along with its bio-compatibility and corrosion
resistance, makes it an attractive material for many medical applications including
stents which are used to counteract the effects associated with vascular diseases, such
as narrowing of blood vessels due to plaque-build-up. A "self-expanding" superelastic stent is a cylindrical metal mesh (typically made by laser-cutting of a tube to
the desired pattern, and subsequently electro-polishing to remove any burrs) which
is compressed to a smaller diameter than its as-manufactured strain-free diameter,
inserted into the problem area in an artery, and allowed to self-expand to its original
diameter while exerting a gentle radial force on the wall of the artery to keep it open.
An electron-micrograph of a Sci-Med"
stent (Serruys, 1997) with struts which have
a square cross-section of 0.15 mm by 0.15 mm is shown in Figure 3-12; this stent has
an expanded diameter of 3.8 mm.
A simple way to experimentally study the mechanics of a stent is to analyze
the response of its struts. Diamond-shaped strut test specimens, Figure 3-13a, were
electro-discharge-machined from our sheet Ti-Ni material. The specimens were deformed in in-plane tension and compression using a special micro-tensile testing device
developed by Gudlavaletti et al. (2002). The forces were measured using a built-in
load cell, while the relative displacements of the end-sections were measured using a
non-contact optical extensometer devised by Su and Anand (2002). To ensure nearisothermal testing conditions, the tests were conducted at a very low loading rate.
The experimentally-measured force-displacement curves in tension and compression
conducted at room temperature (0
=
298K) are shown in Figure 3-14. Note, that
the experiment shows that for the same magnitude of extension in tension and compression, the maximum tensile load is ~ 15 N, while it is ~ -13 N in compression.
The initial finite-element mesh used to model the specimen is shown in Figure 3-13b;
because of the symmetry of the specimen and the loading conditions, only one quarter
of the specimen is modelled. The prediction of the superelastic response based on
the material parameters determined in the previous section for the sheet Ti-Ni is also
96
shown in Figure 3-14. The prediction is in good accord with the experiment. Of particular note is that the experimentally observed tension-compression asymmetry of
the load-displacement response is picked up by the numerical simulation. Figure 3-15
compares numerically-predicted deformed geometries of the specimen in tension and
compression against photographs of the corresponding experimental configurations.
The numerical result also shows the contours of the martensite volume fraction. As
expected, the martensite volume fraction is maximum at the corners where bending
effects are dominant.
Next, to demonstrate the capability of our model to perform complex threedimensional superelasticity simulations in a reasonable time, we report on a numerical
simulation of the deformation response of a Ti-Ni stent using the implementation of
our model in ABAQUS/Standard.9 A 4 mm long link of the Sci-Medt" stent is meshed
using 5,562 ABAQUS-C3D8 elements. The initial finite-element mesh is shown in Figure 3-16a. Representative nodes at the tips of the five V-shaped segments at one end
of the link of the stent are fixed so as not to move in the axial direction, while a
forward-and-reverse axial displacement is prescribed at corresponding nodes at the
other ends of the V-shaped segments. The material parameters in the analysis are
taken to be the same as the ones used in the previous section. Figure 3-16b shows
the fully extended state of the stent together with the contours of martensite volume fraction. As expected, the largest volume fraction of martensite occurs at the
cross-sections where the bending strains are the highest. The final diameter of the
stent in the extended state is about four times smaller than the diameter in the initial
state. Note that because of the geometric design of the stent there is only a relatively
small axial extension ~ 0.16 mm, even for such a large reduction in diameter. The
numerically calculated load-displacement curve is shown in Figure 3-16c; due to its
geometric design the stent stiffens considerably as it is being extended. Of course,
9
Recently, Rebelo and co-workers (e.g., Rebelo and Perry, 2000) have implemented a version of
the superelasticity model of Auricchio and Taylor (1996, 1997) in ABAQUS and used it to simulate
the deformation response of Ti-Ni stents. Details of the precise constitutive model used and its
implementation are not publically known. Simulations of some other medical applications of Ti-Ni
using this computer program have also been recently reported by Gong and Pelton (2002).
97
since the stent material is superelastic, the load returns back to its original position
of zero load at zero extension. This reasonably complex three-dimensional calculation took approximately two hours to complete on a present-day workstation-class
machine.
98
600
1
500
-IMPLICIT
EXPLICIT
1
1
1
1
0.06
0.07
0.08
400
0U
a.
C)
En 300
C,)
200
100
0
0
0.01
0.02
0.03
0.05
0.04
STRAIN
0.09
Figure 3-1: Comparison of the implicit and explicit finite-element solution of the
tension specimen shown in Figure 2-18b at a tensile strain-rate of 0.002/s at an
initial temperature of 6 = 300 K using the material parameters in Section 3.2.1.
99
CY
0 =const.
en
CTr
amI----
I
fc
x
_TT(0-O T)
a
4E
0'
E
(a)
am(0 3 )
T_(0-0OT)
&T (
-T
m(02)
am
Cyam-(01)
,,'
T
&r
-
0 ms
0T
ma 03)
-ma(02)
,mam
0 as
(b)
0
Figure 3-2: (a) Schematic of an isothermal superelastic stress-strain response in simple
tension. (b) Schematic diagram of the variation of the nucleation stresses -am and
Uma with temperature.
100
450
400-
o EXPERIMENT: 0 ; 289 K
350-
- -FIT
300
250
Cl)
150-
0
10
000
100- 1
50
I
~
0
0
0.04
0.03
STRAIN
0.02
0.01
0.06
0.05
(a)
450
400
-
350
*
6
300 -
110
0
250
4GH
100
0
XEIMN
- - F1
-c
C(b)
a
50
0
0
030K
e
0
0.02
0.01
EXPERIMENT: 0;300 K
FIT
0.03
0.04
STRAIN
(b)
0.05
0.06
450
400
~
0
350
0
U0
0
Iia
~200
'o
0
0
w
150
CD21000
0
0
Hs
o
0.1
0
R
.2
o EXPERIMENT: C;306 K
.6
.5
.4
-FIT.3
0~
50
lsoshwn
I
0
300
00
0.01
0.02
0.03
0.04
STRAIN
0.05
0.06
(C)
Figure 3-3: Nominal superelastic stress-strain curves from tension tests on 00 -oriented
specimens. The tests were conducted under isothermal conditions at (a) 289 K, (b)
300 K, and (c) 306 K. The experimental data from these tests were used to estimate the
phase transformation parameters. The curve fit from the finite-element simulations
is also shown.
101
500
450
400
e
350
d
--
_--
-
-b
'~-'
--
b/
. 300
I
-
250
9i
200 -h
-
150
-
r -.--
-
-
-------
100 -I
50
aI
0. j
i
0.01
0.02
0.03
0.04
STRAIN
(a)
(b)
0.05
0.06
0.08
0.07
(g)
(g)
1.00
0.89
0.78
0.67
(C)
0.56
0.44
0.33
(h)
0.22
0.11
0.00
(d)
(i)
(e)
(j)
Figure 3-4: A superelastic finite-element simulation of an experiment conducted under
isothermal conditions at 300 K. Contour plots of the martensite volume fraction are
also shown. The set of contours on the left shows the transformation from austeniteto-martensite during forward transformation, while those on the right show the transformation from martensite-to-austenite during reverse transformation. Note that because of the slight taper in the gage width, the austenite-to-martensite transformation
nucleates at the right grip-end and propagates toward the opposing grip-end.
102
0.09
0.08o
-
0.07-
-
EXPERIMENT :0; 150 MPa
PREDICTION
0.06
-
z
0
-
0.05---
0.04
0o*
-
0'1
-
0
0
to
0-
0.03-
~
0
0.02
0
0I0
0.01
0
270
280
310
300
TEMPERATURE [K]
(a)
290
320
330
340
330
340
0.09
0.08EXPERIMENT: 0; 250 MPa
0
PREDICTION
--
0.070.06
z 0.05-
0
'0
10
0
0
C',
0.040.03-
01
0.02
010
0
0.01
0
270
280
290
I
310
320
300
TEMPERATURE [K]
(b)
Figure 3-5: Strain-temperature cycling experiments conducted on 00-oriented specimens at pre-stress levels of (a) 150 MPa; and (b) 250 MPa. The predictions from the
finite-element simulations are also shown.
103
I
600
o
LArLIiIMLI'JI
:u.uu ~/s
I
0 EXPERIMENT: 0.00003 /s
A EXPERIMENT: 0.00084 /s
SEXPEIMEN :m
U.002e /S
500
-
.,. 30
. --3 I
' :i
6
400
(I)
00/
r0,
300
(I)
200
100
0
-0
00
00
03131--
FEM 0.000 03/s
--- FEM :0.000 84 /s
FEM: 0.002 /s
[3'13
0
0
0.01
0.02
0.03
0.05
0.04
STRAIN
0.06
0.07
0.08
0.09
Figure 3-6: Superelastic stress-strain curves in tension along the rolling direction at
nominal strain rates of 2 x 10-3 /sec, 8.4 x 10-4 /sec, and 3 x 10-5 /sec ("isothermal
condition"). The predictions from the theory are also shown.
104
UJV
I
I
I
I
I
I
I
e,f
-'-
d
500-
-
C
400-
U)
Wl
300-
C/,
200 -
I
.. g
100-
aL
i A
0
0.01
I
0.02
I
0.03
I
I
0.05
0.04
STRAIN
(a)
I
0.06
I
I
0.07
0.08
0.09
(f)
1. 00
0. 89
0. 78
0. 67
0. 56
0. 44
0. 33
0. 22
0.
0. 00
(b)
(c)
(g)
(h)
(d)
(i)
(e)
(i)
Figure 3-7: A superelastic finite-element simulation of an experiment conducted at a
nominal strain rate of 2 x 10-3 /sec. The set of the martensite volume fraction contours on the left show the transformation from austenite-to-martensite during forward
transformation, while those on the right show the transformation from martensite-toaustenite during reverse transformation. Note that because of the boundary conditions, both the forward and reverse transformations nucleate from the grip-ends and
propagate towards the center of the specimen.
105
600
e,f
500
d
c---
C---
b
400
.
0Cl)
300
-9
ICl,
/h
200
i
et
100
ii
i
i
/
, -'o
at
0
0.01
0.02
0.03
0.05
0.04
STRAIN
(a)
0.06
0.07
0.08
0.09
____
(f)
O (K)
(b)
(c)
325.0
320.3
314.7
309.0
303.3
297 .7
292.0
286.3
280.7
275.0
(g)
(h)
(d)
(i)
(e)
(j)
Figure 3-8: A superelastic finite-element simulation of an experiment conducted at
a nominal strain rate of 2 x 10-3 /sec. The set of contours on the left shows the
temperature increase during transformation from austenite-to-martensite, while those
on the right show the temperature decrease during transformation from martensiteto-austenite. During forward transformation the temperature increases by as much
as 25'K above the ambient temperature of 300 0 K due to the exothermic austeniteto-martensite transformation, whereas it decreases by as much as 25 0 K below the
ambient temperature during the reverse endothermic transformation from martensiteto-austenite.
106
0.07
0.06
0.05
z
U)
0.04
0.03
0.02
0.01
0
0
20
40
60
80
TIME [s]
(a)
100
120
140
700
600
o EXPERIMENT
- - PREDICTION
500
c400
9
00 0
U),
Cn
w
(n 300
200
00/I
111'0
-
00
I0
0-
-O
0
1 e6
100
0
-
Of
e
II
-00
0
0
0.01
0.02
0.03
0.04
STRAIN
(b)
0.05
0.06
0.07
6 ~rc~tse
Figure 3-9: (a) Nominal strain versus time profile for the tension-hold experiment at
a baseline nominal strain rate of 1.25 x 10- /sec and initial temperature of 300 K.
(b) Superelastic stress-strain curve measured in the experiment. The prediction from
the finite-element simulation is also shown.
107
f-i
1000
900800-
-
o EXPERIMENT
FIT
700-
C-
600-
F
C',
500C',
400
300
200
100
'V
I
0.04
0.03
STRAIN
0.02
0.01
A
-
- __
t2~7
-
0I
NEEN f-A
I
I
0.07
0.06
0.05
(a)
500
I
I
I
I
a
I
I
450o EXPERIMENT
- PREDICTION
400350300-
Z:
250-
Hn
200-
/
150 -
1001
50
0
0~w01
01
0.02
0.02
0.03
0.03
0.04 0.05 0.06
SHEAR STRAIN
(b)
00pp.
0.07
0.08
0.09
0.1
Figure 3-10: (a) Superelastic stress-strain curve in tension. The experimental data
from this test were used to estimate the constitutive parameters. The curve fit using the constitutive model is also shown, and (b) Superelastic stress-strain curve in
torsion. The prediction from the constitutive model is also shown.
108
--
~
-
-
-
H-Un
-
0.05
Axial strain
Shear strain
0.04
W0.03I
-.
-
0.02
0.01
oh
0
100 200
200
300
300
400 500 600
400 500 600
TIME [s]
MART. VOL. FRAC.
700 800
700 800 900 1000
0 .0 0
0.33
(a)
Km0.44
-0.56
0.78
0.89
1.00
(b)
o
500
-
(C)
350
600
-300
a-
EXPERIMENT
PREDICTION
A
EXPERIMENT
- - PREDICTION
250
(L 400
200
0)
A
AA
AAA AA
W 300
w 150
-
200
00000
0
AA
/AA
C,
1001
00000
100
A
A
4
0
tj
%- -- - -- -- A__4a6
0
0.005
0.01
0.015 0.02
STRAIN
0.025 0.03
0.035 0.04
A
A
0
A/
A
.
50
A
/
-
A
t,
A
A
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
SHEAR STRAIN
(e)
(d)
Figure 3-11: (a) Loading program for combined tension-torsion experiment. (b) Undeformed mesh of 768 ABAQUS C3D8R elements. (c) Deformed mesh at a tensile
strain of 3% and a shear strain of 4.5%. Contours of martensite volume fraction are
shown. (d) Superelastic stress-strain curve in tension. (e) Superelastic stress-strain
curve in shear. The prediction from the constitutive model is also shown.
109
a
b
Figure 3-12: (a) Illustration of the SciMed stent. The 14-mm stent consists of five
segments attached to each other at three sites. This design allows for good support
without gaps, and high flexibility, and (b) Scanning electronic microscopic picture of
the SciMed stent with full expansion. (Taken from the Handbook of coronary stents,
(Serruys, 1997)).
110
1.500
8
11
RO.00
R
10..50
0
R2
(a)
2
3
(b)
Figure 3-13: (a) Geometry of diamond-shaped strut test specimens; all dimensions
are in millimeters. The sheet form which the specimens were machined is 0.53 mm
thick. (b) Initial finite-element mesh of one-quarter of the specimen.
111
I
I
I
i
I
I
15-
10-
5-
0
0-
-5
-
-
-- - - - - - - -
-10 -o
EXPERIMENT
PREDICTION
-15 1
-2.5
-2
-1.5
-1
0.5
0
-0.5
DISPLACEMENT [mm]
1
1
__ I
1.5
2
2.5
Figure 3-14: Superelastic force-displacement curve of the diamond-shaped strut test
specimen in tension and compression. The corresponding prediction from the finiteelement simulation is also shown.
112
MART.
VOL.
FRAC.
0.00
-0 . 11
" li
o .22
-0.33
0. 44
0. 56
0. 67
0.78
0 .89
1. 00
initial position
tension
compression
(a)
(b)
Figure 3-15: (a) The physical position of the diamond-section stent, and (b) the
corresponding finite-element mesh with contours of the martensite volume fraction.
113
MART.
VOL.
FRAC.
0.00
0 .11
0.22
0.33
0 .44
0. 56
-0
. 67
.78
.89
1 . 00
0
0
(b)
(a)
55
I
I
I
I
I
i
I
I
/
,I
~I
50
jI
4540-
ii
ii
II
- - FEM SIMULATION
35 -
'I
/
30 -
II
I,
0
-1
I
II
I
25-
/
/
/
20 -
/
15
10
5
0
'
0.02
0
0.04
0
0.06
0
0.08
I
I
I
0.1
0.12
0.14
-
I
0.16
0.18
DISPLACEMENT [mm]
(c)
Figure 3-16: (a) Initial finite-element mesh of a segment of the stent. (b) The fully
crimped state of the stent with contours of martensite volume fraction. (c) Predicted
axial load-displacement curve from the finite-element simulation.
114
Chapter 4
Conclusion and future work
A crystal-mechanics-based constitutive model to describe the superelastic response
of polycrystalline shape-memory alloys has been developed and implemented in a
finite-element program. It is shown (to within the idealization of no plastic deformation of austenite and of no de-twinning of martensite) to quantitatively predict
the superelastic response of initially-textured polycrystalline Ti-Ni rod and sheet in a
variety of tension, compression, torsion and combined tension-torsion experiments to
good accord. In particular we have shown that crystallographic texture is the main
cause for the anisotropic response in shape-memory materials. We have also showed
the applicability of the Taylor-type model (1938) as a reasonably accurate and computationally inexpensive method for determining the response of textured Ti-Ni in
multi-axial setting.
The non-isothermal and non-homogeneous deformation exhibited by initiallytextured Ti-Ni was also investigated.
Coupled temperature-displacement analysis
to predict the response of non-isothermal superelastic experiments at higher deformation rates were successfully performed. Furthermore the constitutive model is also
able to qualitatively capture the phenomena of nucleation and propagation of transformation fronts under isothermal and non-isothermal conditions. The trend of the
temperature field in the gage section after a loading and reverse loading step for a
specimen conducted under non-isothermal testing conditions were also predicted to
115
good accord.
Some of the future work includes
* As the size of a shape-memory material sample becomes very small (<~ lAm),
in the case of very thin sputter-deposited SMA films, further analysis by James and
Hane (2000) show the possible existence of new types of microstructures such as
the "tent" and "tunnel" structures. This has also been experimentally observed in
very small specimen sizes but not observed in macroscopic sample sizes which we
have investigated here. These additional microstructures have accompanying habitplane normals and transformation directions which can be calculated according to
the theory of James and Hane (2000). This information can be readily incorporated
into the constitutive model in its present form.
* As discussed in Chapter 2, shape-memory materials will be in the fully martensitic
state below
0
mf.
In this state, the inelastic strains due to stressing is caused by
the motion of the twin boundaries between the martensitic plates and within the
martensitic plates itself. This phenomena is called re-orientationand de-twinning,
respectively.
De-twinning can even occur during superelasticity if the material is
stressed significantly after complete austenite to martensite has occurred. Once reorientation and de-twinning is completed an increase of temperature to above its
austenitic finish temperature 9 af will cause complete recovery to back to its parent
state without any residual deformation. This phenomenon is called the shape-memory
effect. A constitutive model that comprehends the re-orientation and de-twinning of
the martensitic variants, and the shape-memory effect needs to developed.
Since there is a significant computational expense in implementing crystalmechanics-based constitutive models for design purposes an isotropic constitutive
model for numerical simulation of thermo-mechanically-coupled superelastic response
of shape-memory materials has been developed and implemented in a finite-element
program. In addition to the standard thermo-elastic material parameters, our simple model contains only four material parameters {ET,
OT, AT,
fc}
to characterize the
austenite +-+ martensite phase transformations, and these parameters are easily determined from isothermal superelastic experiments at a few different temperatures.
116
Representative isothermal and thermo-mechanically coupled superelastic experiments
on polycrystalline Ti-Ni sheet are shown to be predicted with reasonable accuracy by
the constitutive model and computational capability. The superelastic behavior of a
diamond-section stent has also been experimentally and numerically studied.
The simulation capability demonstrated in Section 3, holds the promise of greatly
reducing the time required to design a new device made from a superelastic shapememory material.
The new capability should allow device-designers to relatively
rapidly carry out numerous design iterations, and to computer-test their devices before any actual prototypes are built.
117
Bibliography
" ABAQUS
reference manual. (1999 & 2001). Providence, R.I.
* Abeyaratne, R., & Knowles, J. (1993).
A continuum model for a thermoelastic
solid capable of undergoing phase transitions. Journal of the Mechanics and Physics
of Solids, 41, 541-571.
* Anand, L., & Gurtin, M.E. (2002). Thermal effects in the superelasticity of crystalline shape-memory materials. Submitted to the Journal of the Mechanics and
Physics of Solids.
* Anand, L., & Kothari, M. (1996). A computational procedure for rate-independent
crystal plasticity. Journal of the Mechanics and Physics of Solids, 44, 525-558.
* Auricchio, F., & Taylor, R. (1997). Shape-memory alloys : modelling and numerical simulations of the finite-strain superelastic behavior. Computational Methods in
Applied Mechanics and Engineering, 143, 175-194.
9 Auricchio, F., Taylor, R., & Lubliner, J. (1997). Shape-memory alloys: macromodeling and numerical simulations of the superelastic behavior. ComputationalMethods
in Applied Mechanics and Engineering, 146, 281-312.
9 Ball, J., & James, R.D. (1987).
Fine phase mixtures and minimizers of energy.
Archive of Rational Mechanics and Analysis, 100, 13-52.
9 Baron, T., Levine, M. L., Hayward, V., Bolduc, M., Grant, D. 1994. (November,
Ottawa). A biologically-motivated robot eye system. Proc. 8th Canadian Astronautics and Space Institute (CASI) Annual Conference. pp. 231-240.
* Bekker, A., & Brinson, L. (1997). Temperature-induced phase transformation in
a shape memory alloy : phase diagram based kinetics approach. Journal of the Mechanics of Solids, 45, 949-988.
e Bhattacharya, K., & Kohn, R. (1996). Symmetry, texture and the recoverable strain
of shape-memory polycrystals. Acta Materialia,44, 529-542.
* Bhattacharya, K., & Kohn, R. (1997). Energy minimization and the recoverable
strains of polycrystalline shape-memory alloys. Archive of Rational Mechanics and
Analysis, 139, 99-180.
118
* Boyd, J., & Lagoudas, D. (1994). A thermodynamical constitutive model for the
shape memory effect due to transformation and reorientation. Proceedings of SPIE InternationalSociety of Optical Engineering, 2189, 276-288.
* Boyd, J., & Lagoudas, D. (1996). A thermodynamical model for shape memory
materials. i. The monolithic shape memory alloy. InternationalJournal of Plasticity,
12, 805-842.
e Brill, T., Mittelbach, S., Assmus, W., Mullner, M., & Luthi, B. (1991).
Elastic
properties of Ni-Ti. Journal of Physics : Condensed Matter, 12, 9621-9627.
* Entemeyer, D., Patoor, E., Eberhardt, A., & Berveiller, M. (2000). International
Journal of Plasticity, 16, 1269-1288.
* Gall, K., & Sehitoglu, H. (1999). The role of texture in tension-compression asymmetry in polycrystalline Ni-Ti. InternationalJournal of Plasticity,15, 69-92.
9 Gao, X., Huang, M., & Brinson, L. (2000). A multivariant micromechanical model
for SMAs part 1. crystallographic issues for single crystal model. InternationalJournal of Plasticity, 16, 1345-1369.
* Gong, X. Y., & Pelton, A.R. (2002). ABAQUS analysis on Nitinol medical applications. Proceedings ABAQUS Users' Conference, 1-10.
* Gudlavaletti, S., Gearing, B.P., & Anand, L. (2002). Design and construction of
micro-mechanical testing machines for measuring the bulk and surface properties of
materials at the meso-scale. In preparation.
e Hane, K., & Shield, T. (1999). Microstructure in the cubic to monoclinic transition
in titanium-nickel shape memory alloys. Acta Materialia,47, 2603-2617.
* Inoue, H., Miwa, N., & Inakazu, W. (1996). Texture and shape memory strain in
TiNi alloy sheets. Acta Materialia,46, 4825-4834.
* Ivshin, Y., & Pence, T. (1994). A thermomechanical model for a one variant shape
memory material. Journal of Intelligent Material, Systems and Structures, 5, 455473.
* James, R., & Hane, K. (2000). Martensitic transformations and shape-memory
materials. Acta Materialia,48, 197-222.
* Kallend, J., Kocks, U., Rollett, S., & Wenk, H. (1989).
119
The Preferred Orientation
Package of Los Alamos. Los Alamos National Laboratory (PoPLa), TMS, Warrendale, PA.
9 Knowles, J. (1999).
Stress-induced phase transitions in elastic solids. Computa-
tional Mechanics, 22, 429-436.
* Liang, C., & Rogers, C. (1990).
One-dimensional thermomechanical constitutive
relations for shape memory materials. Journal of Intelligent Materials, Systems and
Structures, 1, 207-234.
e Lim, T., & McDowell, D. (1999).
Mechanical behavior of a Ni-Ti shape memory
alloy under axial-torsional proportional and on-proportional loading. Journal of Engineering Materials and Technology - Transactions of the ASME, 121, 9-18.
e Lu, Z., & Weng, G. (1998). A self-consistent model for the stress-strain behavior
of shape-memory alloy polycrystals. Acta Materialia,46, 5423-5433.
9 Matsumoto, 0., Miyazaki, S., Otsuka, K., & Tamura, H. (1987). Crystallography
of martensitic transformation in Ti-Ni crystals. Acta Metallurgica,35, 2137-2144.
* Miyazaki, S., & Otsuka, K. (1989).
Development of shape memory alloys. ISIJ
International,29, 353-377.
* Otsuka, K., & Wayman, C. (1999). Shape memory materials. Cambridge University Press, Cambridge.
* Patoor, E., Eberhardt, A., & Berveiller, M. (1996). Micromechanical modelling of
superelasticity in shape memory alloys. JournalDe Physique IV, 6, 277-292.
* Qidwai, M. A., & Lagoudas, D. (2000). Numerical implementation of a shapememory alloy thermomechanical constitutive model using return mapping algorithms.
InternationalJournalfor Numerical Methods in Engineering, 47, 1123-1168.
9 Rebelo, N., & Perry, M. (2000). Finite element analysis for the design of Nitinol
medical devices. Minimally Invasive Therapy and Allied Technologies, 9, 75-80.
* Saburi, T., & Nenno, S. (1981). The shape memory effect and related phenomena.
Proceedings of International Conference on Solid-Solid Phase Transformations. The
Metallurgical Society, AIME, New York, 1455-1479.
* Serruys, P. (1997). Handbook of coronary stents. Rotterdam Thoraxcentre Interventional Cardiology Group, Rotterdam.
120
G., & Sung, S. (2001). Private communication.
" Shan,
" Shaw, J., & Kyriakides, S. (1995). Thermomechanical aspects of NiTi. Journal of
the Mechanics and Physics of Solids, 43, 1243-1281.
* Shaw, J., & Kyriakides, S. (1997).
On the nucleation and propagation of phase
transformation fronts in a NiTi alloy. Acta Materialia,45, 683-700.
9 Shield, T. (1995). Orientation dependence of the pseudoelastic behavior of single
crystals of Cu-Al-Ni in tension. Journal of the Mechanics and Physics of Solids, 43,
869-895.
* Shu, Y., & Bhattacharya, K. (1998). The influence of texture on the shape memory
effect in polycrystals. Acta Materialia,46, 5457-5473.
* Sobin, P. (1977). Mechanical properties of Human Veins. M.S. Thesis, University
of California, San Diego, California.
0 Su, C., & Anand, L. (2002). Development of a high resolution optical extensometer
system. In preparation.
* Sun,
Q.,
& Hwang, K. (1993a). Micromechanics modelling for the constitutive be-
havior of polycrystalline shape memory materials - i. Derivation of general relations.
Journal of the Mechanics and Physics of Solids, 41, 1-17.
o Sun,
Q.,
& Hwang, K. (1993b).
Micromechanics modelling for the constitutive
behavior of polycrystalline shape memory materials - ii. Study of the individual phenomena. Journal of the Mechanics and Physics of Solids, 41, 19-33.
o Taylor, G. (1938). Plastic strain in metals. Journal of the Institute of Metals, 62,
30-324.
o Thamburaja, P., & Anand, L. (2001). Polycrystalline shape-memory materials
effect of crystallographic texture. Journal of the Mechanics and Physics of Solids,
49, 709-737.
* Thamburaja, P., & Anand, L. (2002). Superelastic behavior in tension-torsion of an
initially-textured Ti-Ni shape-memory alloy. To appear in the InternationalJournal
of Plasticity.
121
* Zhao, L., Willemse, P., Mulder, J., Beyer, J., & Wei, W. (1998).
Texture
development and transformation strain of a cold-rolled ti50-ni45-cu5 alloy. Scripta
Materialia,39, 1317-1323.
122
Appendix A
Time-integration procedure :
Crystal-mechanics-based
constitutive model
In this appendix we summarize the time-integration procedure that we have used for
our rate-independent single-crystal constitutive model. With t denoting the current
time, At is an infinitesimal time increment, and T
=
t + At, the algorithm is as
follows:
Given: (1) {F(t), F(T), 6(t), 0(r)}; (2) {T(t), FP(t)}; (3) {b',
mulated martensite volume fractions
Calculate: (a){T(T),
FP(T)},
mn,
f,}; (4) the accu-
(t).
(b) the accumulated martensite volume fractions V(T),
and (c) the inelastic work increment AwP(r) and march forward in time.
The steps used in the calculation procedure are:
Step 1. Calculate the trial elastic strain
Ee(T)trial:
Fe (T)trial = F(r)(FP(t))
ce(r)trial
Ee(T)trial
=
=
(Fe (T)trial)T Fe (T)trial,
(1/2)
{Ce (T)trial _ 1}.
123
Step 2. Calculate the trial stress Te(,)trial:
Te (T) t rial = C(t)[Ee(T) t rial
-
A(t)(O(T)
00)].
-
Step 3. Calculate the trial resolved force ri()trial on each transformation system.
The resolved force was defined as
force
Tr(T) =
may be approximated at by
Ti(T)
{Ce(T)Te(r)} - (bi
T(T)
0 m'). The resolved
Te(r) - (b' 0 m') for infinitesimal
elastic stretches. Accordingly, the trial resolved force is calculated as
ri(T)trial = Te(T)trial .
(bi 0 m').
Step 4. Calculate the trial driving force for phase transformation
fi(T)frial = Ti(T)trial
-
(AT/OT) (O(T) -
fi (T)trial:
OT).
Step 5. Determine the set PA of potentially active transformation systems which
satisfy
fi(T)trial _
for the a
> 0,
0
<
'(t) < 1 and 0 < E
'(t) < 1
m transformation, and
-
fi(T)trial
for the m
f
-
+ fc < 0,
0 < i(t) < 1 and 0 < Ei
(t) < 1
a transformation.
Step 6. Calculate
bi 0 mi
FP(T) =A~
FP(t), j = 1, ..., N,
(A.1)
jEPA
where N is the total number of potentially transforming systems. Of the N potentially
active systems in the set PA, only a subset A with elements M < N, may actually be
active (non-zero volume fraction increments). This set is determined in an iterative
fashion described below.
124
During phase transformation, the active systems must satisfy the consistency conditions
(A.2)
f = 0,
fi()
m transformation and the + sign holds during
where the - sign holds during a m -+ a transformation, and where
f'(T) = Tr(T) -
(AT/OT) (0(T) -
(A.3)
OT).
Retaining the terms of first order I in A 3, it is straightforward to show that
7'(T) =
Ti(T) t 'ial -
S (b' ® m') - C(t)[sym(Ce(r)
t
rial(bi ® mi))]a.
(A.4)
jEPA
Use of equations (A.3) and (A.4) in the consistency conditions (A.2) give
I:A
xi
(A.5)
= bi, i E'PA,
jEPA
with
A
(A.6)
fe > 0,
and
x'
> 0
for a -+ m transformation,
(A.7)
= fi(T)trial + fj < 0,
and
x _=AV < 0
for m -* a transformation.
(A.8)
T = fi(T)trial
b
= {(b' 0 mi) . C(t)[sym(Ce(r) t rial(bj 0 mi))]}
-
A
Equation (A.5) is a system of linear equations for the martensite volume fraction
increments xi = Aj.
The following iterative procedure based on the Singular Value Decomposition
(SVD) of the matrix A is used to determine the active transformation systems and
the corresponding martensite volume fraction increments (Anand and Kothari, 1996).
Calculate
x = x+ = A+b,
'Terms such (C"' - Ca) ZjCPA A~j and (A' - A")
Ca)
AA
<< C(t) and (A
m
- Aa) EPA
125
EjEPA A~j are neglected because (C"
j << A(t) for
IA~'I
<< 1.
-
where A+ is the pseudo-inverse matrix of the matrix A; if the matrix A is not singular, then the pseudo-inverse, A+, is the true inverse, A- 1 . If for any system the
solution x = A
< 0 when b' > 0 (during a -> m transformation), then this system
is inactive, and it is removed from the set of potentially active systems PA, and a
new A matrix is calculated. Similarly, if xJ = Aj > 0 when 6' < 0 (during m -- a
transformation), then this system is also inactive, and it is also not included in the
set PA used to determine the new A matrix. This iterative procedure is continued
until all xi = A~j > 0 for a ->
m transformation, and xi = Adi < 0 for m -- a
transformation. The final size of the matrix A is M x M, where M is the number of
active systems in the set A.
Step 7. Update the inelastic deformation gradient FP(r):
FP(T) =
bi 0A
0 m} FP(t).
1+
(A.9)
jE A
Step 8. Update the martensite volume fraction for each system
martensite volume fraction for the single crystal
(
and the total
(T):
(A.10)
(t) + A ',
=
$(Tr)
N
()
(A.11)
(0 ).
=
j=1
If
$
(r) > 1, then set
(r) >
1, then set
$j
(T) =
(T)
=
1 and if j (r) < 0, then set
1 and if
(T)
< 0, then set
(T) =
$j
(T) = 0. Similarly, if
0.
Step 9. Update the effective elastic modulus C(T) and thermal expansion A(r):
C(T) =
{1 -
A(T) = {1 -
(T)} C'
+
(T)C m ,
(r)} A' + (T)Am .
126
Step 10. Compute the elastic deformation gradient Fe(T) and the stress Te(T):
Fe(T) =
(r)(FP(r))
Ce(T) = (Fe(T))
Ee(T)
=
T
Fe (,),
(1/2) {Ce(T)
-
1},
Te(T) = C(r)[Ee(r) - A(r)(9(T) - 00)].
Step 11. Update the Cauchy stress T(T):
T(T)
=
(detF(T))-
{Fe(T)Te(T)(Fe(r))
T
}.
Step 12. Calculate the driving force for phase transformation f2(T) and inelastic
work fraction AwP(T):
T'(T)
-=Te (T) - (b' (D m'),
fi(T) = Tr(r) -
AwP(r)
=
{E
(AT/OT)(O(T) -
f'(7-)A } + (AT/OT)
127
OT),
9(T)A
.
Appendix B
Crystallographic theory of
martensite
This theory neglects any elastic strains and assumes that the martensites have
stretches U and the austenite is unstrained, even in the presence of stresses. These
stretches depend on the constants a, 3, -y and V, which are related to the lattice
constants of the two phases and will be discussed shortly. Since the stretch in the
direction normal to the shearing plane is in general not equal 1.0, the transformation
is not a pure shear. Thus it is not possible to kinematically match a single variant
with stretch U to undeformed austenite. Therefore the analysis of Ball and James
(1987) show that the martensite must then be twinned in order to allow approximate
kinematic compatibility at a planar austenite-martensite interface. This analysis is a
formulation of the crystallographic theory of martensite (CTM).
The alloy of Ti-Ni considered here exhibits a structural phase transformation
from the cubic austenite phase to a low temperature monoclinic martensite phase at
the transformation temperature
0
ms.
There are twelve possible ways to transform
the cubic structure to the monoclinic structure. From Hane and Shield (1999) the
stretches U2 are given by
128
U=
0
p
p
p
o r
,U2=
0
-p
-p
-p
0-
p
pr
U4=
U7=
U10=
-p
0
p
-p
P
a
-r
-p
-T
0-
-p
-P
0
P
-(
-P
or
,
--
-p
U-
-p
0
,U8=
-
a
T
-P
0
-p
-p
or
or
-
p
0
-P
O-r
, U9-
p
P )o(
-p
-p
,U12=
0
pT
T
-p
O-
p
-T
-p
p
-r
)
O
O
OE -T
,ull=
p
T
(
-
p
p
,U6
p
-p
j
p
==U
TT
S)
-p
p
a
U=
0
U
p0
-T
-P
-T
o-
p
-p
p
0
-
These stretches depend on the transformation stretches, a, 3 and y, and the
monoclinic angle, 79. The specific components of the stretches Uj are
0 =
a(a + - sin(79))
a 2 + -y2 + 2aysin(79)
aycos(79)
2
a +
2+
2crysin(9)
I
2
yy + asin(79 ))
a 2 + 72+ 2aysin(P)
2
-y(-y + asin(,9 ))
a 2 + 7y2 + 2aysin(O)
T=(
The transformation stretches are given as a = a/aO,
c/
(v2 ao),
#
=
b/(v' ao), and y =
where the lattice parameter of the cubic cell is ao and the lattice parameters
129
of the monoclinic cell are a, b and c, and V is the angle between the edges with lengths a
and c. The lattice parameter for the austenite phase is measured to be a,
The lattice parameters of the monoclinic phase are measured to be a
=
3.015 A.
2.889 A,
b = 4.120 A, c = 4.622 A and 79 = 96.80'. Therefore the transformation stretches are
a = 0.9582, #3 = 0.9663 and y = 1.0840.
The structure of an austenite-twinned martensite interface is shown in Figure B-1.
Displacements must be continuous across the two types of interfaces(twin boundaries
and the A-M interface) in this structure. Mathematically, this reduces to the statement that the deformation gradients in each region must differ by a rank-one tensor.
That is if FA and FB are the deformation gradients on either side of a planar interface
with normal n (in the reference configuration ) then the difference between these two
deformation gradients must be of the following form,
FA - FB = a 0 n,
(B.1)
where a is the shearing vector. The interface between the two variants of martensite
is a twin boundary and kinematic compatibility across this interface requires,
RABUB - UA
= a on,
(B.2)
where RAB is an orthogonal tensor and represents the relative rotation between the
two twins. The polar decomposition of a deformation gradient, F, into a rotation
times a pure stretch , RU, has also been used. In this equation A and B represent
different choices of the integers {1..12}. The rotation is unknown because the transformation only specifies the stretches. This equation admits three types of solutions
which are known as compound, Type I and Type II twins. These types are defined
by the number of symmetries the crystal structures exhibits across the twin plane.
Compound twins have two planes of symmetry and this gives solutions for a and n
whose components are integers in the cubic basis. However, compound twins only
occur between variants that have shears in the same plane. This results in an average
deformation in the martensite region that has a stretch of 3 in the direction normal
to the shear plane. Thus it is impossible for compound twins to be kinematically
130
compatible with austenite. Therefore, Types I and II solutions can exist and will be
allowed.
Because both of these twins are composed of variants that have shears in different
planes, it is possible to pick a volume fraction, A for variant B (so that variant A has
volume fraction (1 - A)) such that the average deformation in the martensite, given
by
FAB = RH(ARABUB +
(1 - A)RA)
is kinematically compatible with the undeformed austenite.
(B.3)
The rotation of this
average deformation is RH. Requiring kinematical compatibility between the average
martensite deformation and the austenite is termed approximate compatibility. It is
useful to use B.2 to write
FAB
=
RH(UA +
Aa 0 n).
(B.4)
The requirement that the twinned martensite be kinematically compatible with
the austenite can then be written,
FAB-
I = b &m
(B.5)
where I is the identity matrix, which is the deformation gradient in the undeformed
austenite. In B.5 b and m are the shearing and normal vectors for the A-M interface,
respectively.
A solution to the crystallographic theory of martensite (CTM) problem is the
quantities a, n, b, m, RAB, RH and A such that B.2 and B.5 are satisfied for a given
choice of A and B. If a matrix C, is known to be of the form
C = (I + n'®a')(I + a' 0 n')
(B.6)
and it is positive definite with a middle eigenvalue of 1 then Proposition 4 of Ball and
James (1987) gives the vectors a' and n' to be
a' = p
A3(1-
A)
ei +
131
(A
3 - 1) e3
(B. 7)
and
n' = p-1
where A, < (A2
=
3
1)
A3
-VA
(-
1 e 1+ K
A-
(B.8)
l-1 e3),
are the eigenvalues of C with the corresponding el,
e 2 and e 3 . The constant p represents an invariant scaling of the solution and will be
used to chose unit normal vectors. The constant n can take on the values ±1. The
procedure is to solve B.2 first and then solve B.5 by constructing matrices of the form
B.6. Equations B.7 and B.8 then give explicit solutions.
To solve for the interface between the martensite twins, we form the matrix
C
=
U-'U2 U-.
(B.9)
If B.2 is satisfied then C has the form B.6 and following identifications are made,
a = a',
(B.10)
n =UAn'.
(B.11)
and
In this case p is chosen such that n is a unit vector. the two solutions B.7 and B.8
provide to B.2 represent Type I and Type II twins. Once B.7, B.8, B.10 and B.11
are used to find a and n then the rotation RAB is found from B.2 directly for a given
choice of r,. The value of K used in B.7 and B.8 for the solution to B.2 will be called
i1.
Once the twin interface is known then we can proceed to apply Proposition 4 of
Ball and James (1987) to determine the A-M orientation. In this case the matrix
CO(A), given by
Co(A)
= (UA +
An ® a)(UA + Aa ® n),
(B.12)
is of the form
C = (I+ m 0 b)(I + b 0 m)
if kinematic compatibility as represented by B.6 is satisfied.
(B.13)
Here A needs to be
solved. Propositions 5 and 6 of Ball and James (1987) state that a matrix of the form
132
B. 13 has eigenvalues that satisfy the requirement of Proposition 4 of Ball and James
(1987) if the value of A is root of the function
g(A) = det(Co(A) - I).
(B.14)
Simplifying B.14 yields to equation 5.56 in Ball and James (1987),
g(A)
=
det1(U
-I) + (A2 - A)(ja 21
-
2
1 n12 ).
(det U2) U1a1
IU-
(B.15)
Once the roots of g(A) are founds then the application of Proposition 4 proceeds
as above and
b b-(
= p (
A(1 -A 1 ) el + K
A(I
A -A,
3
m- =X-:
-A-
1, (A
A3
- 1) e3
(B.
B. 16)
A3 - A,
(- /1 - A, el + K x/h -
e3).
(B. 17)
Note that the solution of A must be in [0,1]. Equations B.3 and B.5 are then used
to determine RH. Again the value of K in B.7 and B.8 determines which of the two
solutions to B.5 for a given value of A is found. The value of K used in B.7 and B.8
when solving B.5 will be called
'2.
Thus there are two roots of g(A) in [0,1] and each
root results in two solutions corresponding to the two values of K2 . Thus, there are
four A-M interfaces possible for a given martensite twin. There are 24 choices for A
and B (omitting the compound twin case) and each pair can form either a Type I
or Type II twinned martensite corresponding to the two values of K1. Thus we have
24 x 2 x 4 = 192 A-M interfaces possible in this material.
133
Austenite
A-M interface (habit plane)
twin boundariesA
B
A
B
m
A
B
b
Twinned Martensite
(Covariant Pair)
A,B -- Martensite Variants
(a)
m
Austenite
b
A
-
n
aB
A-M interface
B
Twinned-Martensite
(b)
Figure B-1: (a) Schematic diagram of an austenite-twinned martensite transformation
system, and (b) a magnified schematic diagram of the twinned martensite.
134
Appendix C
Algorithm to calculate the
austenite-martensite
transformation systems using the
crystallographic theory of
martensite
In this appendix we summarize the algorithm to calculate the austenite-martensite
transformation systems using the crystallographic theory of martensite. For further
details refer to Ball and James (1987) and Hane and Shield (2001).
Given: {UA, UB, K1
= ±1, K2 =
±1}.
Calculate: {a, n, b,m, A, RAB, RH}The steps used in the calculation procedure are:
Step 1. Construct the CAB matrix:
CAB
=UA
UB AU.
Step 2. Perform spectral decomposition of CAB to calculate its eigenvalues A1,
with A, < A2
=
1 < A3 , and eigenvectors ej :
CAB
A, el 0 e 1 + A2 e 2 9 e 2 + A3 e 3 0 e 3 .
135
Step 3. Calculate n' and a':
n
=
//A
FA(A3-I-
,7
(- /I -
i
ei+ -,
I e3),
A1)e +
,(A3 - 1) e
A,
A3~A,
Step 4. Calculate p and scale n' and a' to choose unit normal vectors
p= In',
n'=p n'
a'= pa'.
and
Step 5. Calculate n and a:
n = UA n',
a'.
Step 6. Using equation 5.56 from Ball and James (1987) construct the quadratic
function g(A) and calculate the roots A, and A2 of g(A):
g(A) = det (U2 - 1) + (A2 - A) (Jat 2 - (det U2) IU-' a 2 IU- n12 ).
Step 7. Construct the C(Aj) and C(A2 ) matrices :
C(AI) = (UA+ A, n 9 a) (UA+
A, a 0 n),
C(A2 ) = (UA+ A2 n 0 a)(UA+ A2 aO n).
Step 8. Perform spectral decomposition of C(AI) to calculate its eigenvalues A1 ,
with A
A2 =
1 < A 3 , and eigenvectors ej :
C(AI) = A, el 0 el + A2 e 2
e 2 + A3 e 3
e3 .
Step 9. Calculate m and b:
(m(
A(1 -)
b= (F3A
-A, e1 +
1)
_
el+K2
136
(__
A,3
A2N
3-1l e3 ),
-_ 1)
e3
Step 10. Calculate p and scale m and b to choose unit normal vectors
p=Im,
m=p' m
and
b=pb.
Step 11. Calculate the relative rotation RAB:
RAB = (UA
1
+ A2 aon)UB
Step 12. Calculate the rotation RH of the average austenite-martensite deformation:
RH=
(1+ b 0 m) (UA+ A, a 0 n)-1 .
Repeat steps 8-12 for the C(A2 ) matrix.
C.1
Results of the CTM theory
The crystallographic theory of martensite (CTM) can be used to calculate the transformation and twinning systems for different alloy systems. For example TitaniumNickel (Ti-Ni) alloys undergo a cubic to monoclinic transformation. Copper-based
alloys such as Cu-Al-Ni undergo a cubic to orthorhombic transformation whereas
Indium-Thalium (In-Th) alloys undergo a cubic to tetragonal transformation.
Shield (1995) has calculated the transformation systems for Cu-Al-Ni systems
undergoing cubic to orthorhombic transformation. For Cu-Al-Ni undergoing cubic
to orthorhombic transformation there are six different variants i.e U 1 ..U 6 . Corresponding to these six different variants are a possibility of 96 different transformation
systems. These transformation systems are documented in Shield (1995).
Transformation systems for Ti-Ni alloys have also been calculated by Hane and
Shield (1999). The resulting 192 transformation systems for Ti-Ni alloys are calculated here using the CTM theory and the results are summarized from Tables 2 to
25. Each table has 8 possible transformation systems for a particular variant pairing. The 24 transformation systems which are experimentally observed and used in
our calculations are labelled with an asterisk (*). The 192 possible transformation
systems for Ti-Ni alloys are :
137
Table 2. UA= U 1 and UB = U 3
rno,1
bo,2
MO, 2
rnO,3
bo,2
-0.9087
-0.3334 0.2514 0.0527 -0.0535
0.3334 0.2514 0.0527 0.0535
-0.9087
0.4334 -0.8320 0.1211 0.0397
-0.3464
-0.4334
-0.3464
-0.8320 0.1211 -0.0397
* -0.8888 -0.4045 0.2153 0.0568 -0.0638
0.4045 0.2153 0.0568 0.0638
* -0.8888
0.5137 -0.7711 0.1195 0.0485
-0.3762
-0.5137 -0.7711 0.1195 -0.0485
-0.3762
bo,3
0.1060
0.1060
-0.0257
-0.0257
0.0991
0.0991
-0.0216
-0.0216
Table 3. UA = U 1 and UB = U 4
Mo,1
rnO,2
rnO,3
bo, 1
bO, 2
bo,3
* -0.8888
* -0.8888
-0.3762
-0.3762
-0.9087
-0.9087
-0.3464
-0.3464
0.2153
0.2153
-0.7711
-0.7711
0.2514
0.2514
-0.8320
-0.8320
-0.4045
0.4045
0.5137
-0.5137
-0.3334
0.3334
0.4334
-0.4334
0.0568
0.0568
0.1195
0.1195
0.0527
0.0527
0.1211
0.1211
0.0991
0.0991
-0.0216
-0.0216
0.1060
0.1060
-0.0257
-0.0257
-0.0638
0.0638
0.0485
-0.0485
-0.0535
0.0535
0.0397
-0.0397
Table 4. UA
=
U 2 and UB = U 3
mo,1
nMO, 2
rMO,3
bo,1
bo,2
bo, 3
-0.3762
-0.3762
* -0.8888
* -0.8888
-0.3464
-0.3464
-0.9087
-0.9087
0.7711
0.7711
-0.2153
-0.2153
0.8320
0.8320
-0.2514
-0.2514
-0.5137
0.5137
0.4045
-0.4045
-0.4334
0.4334
0.3334
-0.3334
0.1195
0.1195
0.0568
0.0568
0.1211
0.1211
0.0527
0.0527
0.0216
0.0216
-0.0991
-0.0991
0.0257
0.0257
-0.1060
-0.1060
-0.0485
0.0485
0.0638
-0.0638
-0.0397
0.0397
0.0535
-0.0535
138
Table 5. UA
rno,1
-0.3762
-0.3762
* -0.8888
* -0.8888
-0.3464
-0.3464
-0.9087
-0.9087
MO, 2
-0.5137
0.5137
0.4045
-0.4045
-0.4334
0.4334
0.3334
-0.3334
U 2 and UB = U 4
rMO,3
bo,2
bo, 1
=
0.7711
0.7711
-0.2153
-0.2153
0.8320
0.8320
-0.2514
-0.2514
0.1195
0.1195
0.0568
0.0568
0.1211
0.1211
0.0527
0.0527
Table 6. UA = U 5 and UB
m, 1
-0.3334
0.3334
0.4334
-0.4334
0.5137
* 0.4045
* -0.4045
-0.5137
=
bo,3
0.0216
0.0216
-0.0991
-0.0991
0.0257
0.0257
-0.1060
-0.1060
U7
rMO, 2
MO,3
bo, 1
bo,2
bo,3
-0.9087
-0.9087
-0.3464
-0.3464
-0.3762
-0.8888
-0.8888
-0.3762
0.2514
0.2514
-0.8320
-0.8320
-0.7711
0.2153
0.2153
-0.7711
-0.0535
0.0535
0.0397
-0.0397
0.0485
0.0638
-0.0638
-0.0485
0.0527
0.0527
0.1211
0.1211
0.1195
0.0568
0.0568
0.1195
0.1060
0.1060
-0.0257
-0.0257
-0.0216
0.0991
0.0991
-0.0216
Table 7. UA
=
U 5 and UB = U 8
Mn, 1
nMO, 2
MO,3
0.2153
0.2153
-0.7711
-0.7711
0.2514
0.2514
-0.8320
-0.8320
-0.8888
-0.8888
-0.3762
-0.3762
-0.9087
-0.9087
-0.3464
-0.3464
-0.4045
0.4045
0.5137
-0.5137
-0.3334
0.3334
0.4334
-0.4334
*
*
-0.0485
0.0485
0.0638
-0.0638
-0.0397
0.0397
0.0535
-0.0535
139
bo, 1
0.0991
0.0991
-0.0216
-0.0216
0.1060
0.1060
-0.0257
-0.0257
bo, 2
bo,3
0.0568
0.0568
0.1195
0.1195
0.0527
0.0527
0.1211
0.1211
-0.0638
0.0638
0.0485
-0.0485
-0.0535
0.0535
0.0397
-0.0397
Table 8. UA
=
U 6 and UB
Mo, 1
rnO,2
MO, 3
0.7711
0.7711
* -0.2153
* -0.2153
0.8320
0.8320
-0.2514
-0.2514
-0.3762
-0.3762
-0.8888
-0.8888
-0.3464
-0.3464
-0.9087
-0.9087
-0.5137
0.5137
0.4045
-0.4045
-0.4334
0.4334
0.3334
-0.3334
Table 9. UA
=
=
b0 2
bo,
bo,3
0.1195
0.1195
0.0568
0.0568
0.1211
0.1211
0.0527
0.0527
-0.0485
0.0485
0.0638
-0.0638
-0.0397
0.0397
0.0535
-0.0535
,
0.0216
0.0216
-0.0991
-0.0991
0.0257
0.0257
-0.1060
-0.1060
U 6 and UB
U7
=
U8
rMo, 1
rnO,2
rMO,3
bo, 1
bo, 2
bo,3
0.4045
0.5137
-0.5137
* -0.4045
-0.4334
0.4334
0.3334
-0.3334
-0.8888
-0.3762
-0.3762
-0.8888
-0.3464
-0.3464
-0.9087
-0.9087
-0.2153
0.7711
0.7711
-0.2153
0.8320
0.8320
-0.2514
-0.2514
0.0638
0.0485
-0.0485
-0.0638
-0.0397
0.0397
0.0535
-0.0535
0.0568
0.1195
0.1195
0.0568
0.1211
0.1211
0.0527
0.0527
-0.0991
0.0216
0.0216
-0.0991
0.0257
0.0257
-0.1060
-0.1060
*
Table 10. UA
=
U9 and UB
=
U11
Mo, 1
nO,2
rMO,3
bo, 1
bo, 2
bo,3
-0.7711
* 0.2153
* 0.2153
-0.7711
0.2514
0.2514
-0.8320
-0.8320
0.5137
0.4045
-0.4045
-0.5137
-0.3334
0.3334
0.4334
-0.4334
-0.3762
-0.8888
-0.8888
-0.3762
-0.9087
-0.9087
-0.3464
-0.3464
-0.0216
0.0991
0.0991
-0.0216
0.1060
0.1060
-0.0257
-0.0257
0.0485
0.0638
-0.0638
-0.0485
-0.0535
0.0535
0.0397
-0.0397
0.1195
0.0568
0.0568
0.1195
0.0527
0.0527
0.1211
0.1211
140
Table 11. UA = U9 and UB
=
U 12
no,1
MO, 2
rnO,3
bo,1
bo,2
bo,3
0.5137
0.4045
-0.4045
-0.5137
-0.3334
0.3334
0.4334
-0.4334
-0.7711
0.2153
0.2153
-0.7711
0.2514
0.2514
-0.8320
-0.8320
-0.3762
-0.8888
-0.8888
-0.3762
-0.9087
-0.9087
-0.3464
-0.3464
0.0485
0.0638
-0.0638
-0.0485
-0.0535
0.0535
0.0397
-0.0397
-0.0216
0.0991
0.0991
-0.0216
0.1060
0.1060
-0.0257
-0.0257
0.1195
0.0568
0.0568
0.1195
0.0527
0.0527
0.1211
0.1211
*
*
Table 12. UA = U1 o and UB
=
U11
mo, 1
MO, 2
MO,3
bo, 1
bo, 2
bo,3
0.4045
0.5137
-0.5137
* -0.4045
-0.4334
0.4334
0.3334
-0.3334
-0.2153
0.7711
0.7711
-0.2153
0.8320
0.8320
-0.2514
-0.2514
-0.8888
-0.3762
-0.3762
-0.8888
-0.3464
-0.3464
-0.9087
-0.9087
0.0638
0.0485
-0.0485
-0.0638
-0.0397
0.0397
0.0535
-0.0535
-0.0991
0.0216
0.0216
-0.0991
0.0257
0.0257
-0.1060
-0.1060
0.0568
0.1195
0.1195
0.0568
0.1211
0.1211
0.0527
0.0527
*
Table 13. UA = U 1 o and UB = U 12
rno,1
MO, 2
rnO,3
bo, 1
bo, 2
bo,3
-0.2153
0.7711
0.7711
* -0.2153
0.8320
0.8320
-0.2514
-0.2514
0.4045
0.5137
-0.5137
-0.4045
-0.4334
0.4334
0.3334
-0.3334
-0.8888
-0.3762
-0.3762
-0.8888
-0.3464
-0.3464
-0.9087
-0.9087
-0.0991
0.0216
0.0216
-0.0991
0.0257
0.0257
-0.1060
-0.1060
0.0638
0.0485
-0.0485
-0.0638
-0.0397
0.0397
0.0535
-0.0535
0.0568
0.1195
0.1195
0.0568
0.1211
0.1211
0.0527
0.0527
*
141
rn, 1
-0.0008
-0.8138
-0.8589
-0.2779
0.0419
-0.8887
-0.9098
-0.2012
Table 14. UA = U1 and UB = U 5
bo,1
bo,2
MO, 3
rnO,2
-0.8138 0.5811 0.0934 0.0350
-0.0008 0.5811 0.0350 0.0934
-0.2779 -0.4302 0.0052 0.0901
-0.8589 -0.4302 0.0901 0.0052
-0.8887 0.4565 0.1008 0.0278
0.4565 0.0278 0.1008
0.0419
-0.2012 -0.3629 0.0010 0.0999
-0.9098 -0.3629 0.0999 0.0010
bo,3
0.0433
0.0433
-0.0607
-0.0607
0.0375
0.0375
-0.0485
-0.0485
Table 15. UA = U 1 and UB = U 9
mo,1
MO, 2
MO,3
bo,1
bo, 2
bo,3
-0.0008
-0.8138
-0.8589
-0.2779
0.0419
-0.8887
-0.9098
-0.2012
0.5811
0.5811
-0.4302
-0.4302
0.4565
0.4565
-0.3629
-0.3629
-0.8138
-0.0008
-0.2779
-0.8589
-0.8887
0.0419
-0.2012
-0.9098
0.0934
0.0350
0.0052
0.0901
0.1008
0.0278
0.0010
0.0999
0.0433
0.0433
-0.0607
-0.0607
0.0375
0.0375
-0.0485
-0.0485
0.0350
0.0934
0.0901
0.0052
0.0278
0.1008
0.0999
0.0010
Table 16. UA
=
U 2 and UB = U 7
Mo, 1
nMO, 2
MO, 3
bo, 1
bo, 2
bo,3
0.9098
-0.2012
-0.0419
-0.8887
0.8589
-0.2779
0.0008
-0.8138
-0.2012
0.9098
-0.8887
-0.0419
-0.2779
0.8589
-0.8138
0.0008
-0.3629
0.3629
0.4565
-0.4565
-0.4302
0.4302
0.5811
-0.5811
-0.0010
0.0999
-0.1008
0.0278
-0.0052
0.0901
-0.0934
0.0350
0.0999
-0.0010
0.0278
-0.1008
0.0901
-0.0052
0.0350
-0.0934
-0.0485
0.0485
0.0375
-0.0375
-0.0607
0.0607
0.0433
-0.0433
142
mo,1
-0.0419
-0.2012
0.9098
-0.8887
0.0008
-0.2779
0.8589
-0.8138
Table 17. UA = U 2 and UB = U 1 2
bo,2
bo,1
rnO,3
MO, 2
0.4565
0.3629
-0.3629
-0.4565
0.5811
0.4302
-0.4302
-0.5811
-0.8887
0.9098
-0.2012
-0.0419
-0.8138
0.8589
-0.2779
0.0008
Table 18. UA
=
-0.1008
0.0999
-0.0010
0.0278
-0.0934
0.0901
-0.0052
0.0350
0.0375
0.0485
-0.0485
-0.0375
0.0433
0.0607
-0.0607
-0.0433
U 3 and UB
=
bo,3
0.0278
-0.0010
0.0999
-0.1008
0.0350
-0.0052
0.0901
-0.0934
U6
M ,1
rnO,2
rmo,3
bo, 1
bo, 2
bo,3
0.8589
-0.8138
0.0008
-0.2779
0.9098
-0.2012
-0.0419
-0.8887
-0.2779
0.0008
-0.8138
0.8589
-0.2012
0.9098
-0.8887
-0.0419
0.4302
0.5811
-0.5811
-0.4302
0.3629
-0.3629
-0.4565
0.4565
-0.0052
0.0350
-0.0934
0.0901
-0.0010
0.0999
-0.1008
0.0278
0.0901
-0.0934
0.0350
-0.0052
0.0999
-0.0010
0.0278
-0.1008
0.0607
0.0433
-0.0433
-0.0607
0.0485
-0.0485
-0.0375
0.0375
mr, 1
-0.8589
-0.2779
-0.0008
-0.8138
-0.9098
-0.8887
0.0419
-0.2012
Table 19. UA = U 3 and UB = U 1 1
MO, 2
rMO,3
bo,1
bo,2
0.4302
0.4302
-0.5811
-0.5811
0.3629
-0.4565
-0.4565
0.3629
-0.2779
-0.8589
-0.8138
-0.0008
-0.2012
0.0419
-0.8887
-0.9098
143
0.0052 0.0607
0.0901 0.0607
0.0934 -0.0433
0.0350 -0.0433
0.0010 0.0485
0.0278 -0.0375
0.1008 -0.0375
0.0999 0.0485
bo,3
0.0901
0.0052
0.0350
0.0934
0.0999
0.1008
0.0278
0.0010
Table 20. UA
=
rno, 1
rnO,2
rnO,3
0.0419
-0.8887
-0.9098
-0.2012
-0.0008
-0.2779
-0.8589
-0.8138
-0.8887
0.0419
-0.2012
-0.9098
-0.8138
-0.8589
-0.2779
-0.0008
-0.4565
-0.4565
0.3629
0.3629
-0.5811
0.4302
0.4302
-0.5811
Table 21. UA
no,1
0.9098
-0.2012
-0.0419
-0.8887
0.8589
-0.8138
0.0008
-0.2779
mr, 1
0.5811
0.5811
-0.4302
-0.4302
0.4565
0.4565
-0.3629
-0.3629
MO, 2
0.3629
-0.3629
-0.4565
0.4565
0.4302
0.5811
-0.5811
-0.4302
=
U 4 and UB = U 8
bo,1
bo,2
0.1008
0.0278
0.0010
0.0999
0.0934
0.0901
0.0052
0.0350
0.0278
0.1008
0.0999
0.0010
0.0350
0.0052
0.0901
0.0934
U 4 and UB = U 10
rnO,3
bo, 1
-0.2012 -0.0010
0.9098 0.0999
-0.8887 -0.1008
-0.0419 0.0278
-0.2779 -0.0052
0.0008 0.0350
-0.8138 -0.0934
0.8589 0.0901
bo, 2
bo,3
0.0485
-0.0485
-0.0375
0.0375
0.0607
0.0433
-0.0433
-0.0607
0.0999
-0.0010
0.0278
-0.1008
0.0901
-0.0934
0.0350
-0.0052
Table 22. UA = U 5 and UB = U 9
MO, 2
MO, 3
bo, 1
bo, 2
-0.0008
-0.8138
-0.8589
-0.2779
0.0419
-0.8887
-0.9098
-0.2012
bo,3
-0.0375
-0.0375
0.0485
0.0485
-0.0433
0.0607
0.0607
-0.0433
-0.8138
-0.0008
-0.2779
-0.8589
-0.8887
0.0419
-0.2012
-0.9098
144
0.0433
0.0433
-0.0607
-0.0607
0.0375
0.0375
-0.0485
-0.0485
0.0934
0.0350
0.0052
0.0901
0.1008
0.0278
0.0010
0.0999
bo,3
0.0350
0.0934
0.0901
0.0052
0.0278
0.1008
0.0999
0.0010
Table 23. UA
m, 1
0.4565
0.3629
-0.3629
-0.4565
0.5811
0.4302
-0.4302
-0.5811
nO, 2
nO,3
-0.0419
-0.2012
0.9098
-0.8887
0.0008
-0.2779
0.8589
-0.8138
-0.8887
0.9098
-0.2012
-0.0419
-0.8138
0.8589
-0.2779
0.0008
Table 24. UA
no,1
0.3629
0.3629
-0.4565
-0.4565
0.4302
0.4302
-0.5811
-0.5811
1
0.4302
0.5811
-0.5811
-0.4302
0.3629
0.4565
-0.4565
-0.3629
U 6 and UB
bo, 1
0.0375
0.0485
-0.0485
-0.0375
0.0433
0.0607
-0.0607
-0.0433
=
=
U11
bo,2
-0.1008
0.0999
-0.0010
0.0278
-0.0934
0.0901
-0.0052
0.0350
U7 and UB
=
bo, 3
0.0278
-0.0010
0.0999
-0.1008
0.0350
-0.0052
0.0901
-0.0934
U 12
MO, 2
rnO,3
bo,
bo,2
bo, 3
-0.9098
-0.2012
0.0419
-0.8887
-0.8589
-0.2779
-0.0008
-0.8138
-0.2012
-0.9098
-0.8887
0.0419
-0.2779
-0.8589
-0.8138
-0.0008
0.0485
0.0485
-0.0375
-0.0375
0.0607
0.0607
-0.0433
-0.0433
0.0010
0.0999
0.1008
0.0278
0.0052
0.0901
0.0934
0.0350
0.0999
0.0010
0.0278
0.1008
0.0901
0.0052
0.0350
0.0934
Table 25. UA
rMn,
=
=
U 8 and UB = U 1 0
rnO,2
rnO,3
bo,1
bo, 2
bo,3
0.8589
-0.8138
0.0008
-0.2779
0.9098
-0.8887
-0.0419
-0.2012
-0.2779
0.0008
-0.8138
0.8589
-0.2012
-0.0419
-0.8887
0.9098
0.0607
0.0433
-0.0433
-0.0607
0.0485
0.0375
-0.0375
-0.0485
-0.0052
0.0350
-0.0934
0.0901
-0.0010
0.0278
-0.1008
0.0999
0.0901
-0.0934
0.0350
-0.0052
0.0999
-0.1008
0.0278
-0.0010
145
Appendix D
Example of a fortran code to
calculate the austenite-martensite
transformation systems using CTM
PROGRAM MAIN
C+***** ****
****
**********
**** *** ** ******
*** *
****************
C
THIS PROGRAM CALCULATES THE A USTENITE/MARTENSITE
C
SYSTEMS FOR TI-NI
** ******
C
** **
**********
** *** * ***
*** ***** *** *** ***********
IMPLICIT REAL*8(A-H,O-Z)
PARAMETER(THETA = 0.9563DO, SIGMA
+
RHO = -0.04266DO
=
1.0243DO,TAU = 0.05803DO,
)
REAL*8 Ul(3,3),U2(3,3),U3(3,3),U4(3,3),U5(3,3),U6(3,3),
+
U7(3,3),U8(3,3),U9(3,3),U1O(3,3),U11(3,3),U12(3,3),
+
UA(3,3),UB(3,3),AVEC(3),NVEC(3),UAINV(3,3),
+
BVEC1(3),MVEC1(3),BVEC2(3),MVEC2(3),NTILDE(3)
146
Open file for writing output
cc
OPEN (UNIT=21,FILE=' systems. dat',STATUS='UNKNOWN')
WRITE(21,*) 'RESULTS FROM CRYSTALLOGRAPHIC THEORY OF MARTENSITE'
WRITE(21,*)
'--------------------------------------------------
WRITE(21,*)' 192'
WRITE(21,*)' B(1)
U1(1,1)
=
B(2)
B(3)
THETA
U1(1,2) = RHO
U1(1,3) = RHO
U1(2,2)
=
SIGMA
U1(2,3) = TAU
U1(3,3)
=
SIGMA
U1(2,1)
=
U1(1,2)
U1(3,1)
=
U1(1,3)
U1(3,2)
=
U1(2,3)
U2(1,1) = THETA
U2(1,2) = -RHO
U2(1,3) = -RHO
U2(2,2) = SIGMA
U2(2,3) = TAU
U2(3,3) = SIGMA
U2(2,1) = U2(1,2)
U2(3,1)
=
U2(1,3)
U2(3,2)
=
U2(2,3)
U3(1,1)
=
THETA
U3(1,2)
=
-RHO
147
M(1)
M(2)
M(3)'
U3(1,3) = RHO
U3(2,2) = SIGMA
U3(2,3) =
-TAU
U3(3,3) =
SIGMA
U3(2,1) = U3(1,2)
U3(3,1) = U3(1,3)
U3(3,2) = U3(2,3)
U4(1,1) = THETA
U4(1,2) = RHO
U4(1,3)
=
-RHO
U4(2,2) = SIGMA
U4(2,3) = -TAU
U4(3,3) = SIGMA
U4(2,1) = U4(1,2)
U4(3,1) = U4(1,3)
U4(3,2) = U4(2,3)
U5(1,1) = SIGMA
U5(1,2) = RHO
U5(1,3) = TAU
U5(2,2) = THETA
U5(2,3) = RHO
U5(3,3) = SIGMA
U5(2,1) = U5(1,2)
U5(3,1) = U5(1,3)
U5(3,2) = U5(2,3)
U6(1,1) = SIGMA
U6(1,2) = -RHO
148
U6(1,3) = TAU
U6(2,2) = THETA
U6(2,3) = -RHO
U6(3,3) = SIGMA
U6(2,1) = U6(1,2)
U6(3,1) = U6(1,3)
U6(3,2) = U6(2,3)
U7(1,1) = SIGMA
U7(1,2) = -RHO
U7(1,3)
=
-TAU
U7(2,2) = THETA
U7(2,3) = RHO
U7(3,3) = SIGMA
U7(2,1) = U7(1,2)
U7(3,1) = U7(1,3)
U7(3,2) = U7(2,3)
U8(1,1) = SIGMA
U8(1,2) =RHO
U8(1,3) = -TAU
U8(2,2) = THETA
U8(2,3) = -RHO
U8(3,3) = SIGMA
U8(2,1) = U8(1,2)
U8(3,1) = U8(1,3)
U8(3,2) = U8(2,3)
U9(1,1) = SIGMA
U9(1,2) =TAU
149
U9(1,3) = RHO
U9(2,2)
=
SIGMA
U9(2,3) = RHO
U9(3,3) = THETA
U9(2,1)
=
U9(1,2)
U9(3,1)
=
U9(1,3)
U9(3,2)
=
U9(2,3)
U10(1,1)
= SIGMA
U10(1,2)
=
TAU
U10(1,3) = -RHO
U10(2,2)
=
SIGMA
U10(2,3) = -RHO
U10(3,3)
=
THETA
U10(2,1) = U10(1,2)
U10(3,1) = U10(1,3)
U10(3,2) = U10(2,3)
U11(1,1) = SIGMA
U11(1,2) = -TAU
U11(1,3) = RHO
U11(2,2) = SIGMA
U11(2,3) = -RHO
U11(3,3) = THETA
U11(2,1) = U11(1,2)
U11(3,1) = U11(1,3)
U11(3,2) = U11(2,3)
U12(1,1) = SIGMA
U12(1,2) = -TAU
150
U12(1,3) = -RHO
U12(2,2) = SIGMA
U12(2,3) = RHO
U12(3,3) = THETA
U12(2,1) = U12(1,2)
U12(3,1) = U12(1,3)
U12(3,2) = U12(2,3)
(Example using variant 1 and variant 3)
C
PAIR 1
C
REPEAT FOR ALL OTHER VARIANT COMBINATIONS (Appendix C)
PAIR 1'
WRITE(20,*) '-------------------------C
C
SELECT THE MATRICES UA AND UB
C
DO 10 I=1,3
DO 10 J=1,3
UA(I, J) = U1(I, J)
UB(I, J) = U3(I, J)
10
CONTINUE
C--------------------------------------------------AKAPPA1 = 1.DO
AKAPPA2 = 1.D0
CALL ZEROV(NVEC,3)
CALL ZEROV(AVEC,3)
CALL ZEROV(MVEC1,3)
CALL ZEROV(BVEC1,3)
CALL ZEROV(MVEC2,3)
151
CALL ZEROV(BVEC2,3)
CALL COMPUTE(UA,UB,AKAPPA1,AKAPPA2,NVEC,AVEC,
+
ROOT1,ROOT2,MVEC1,BVEC1,MVEC2,BVEC2)
C--------------------------------------------------AKAPPA1 = 1.DO
AKAPPA2 = -1.DO
CALL ZEROV(NVEC,3)
CALL ZEROV(AVEC,3)
CALL ZEROV(MVEC1,3)
CALL ZEROV(BVEC1,3)
CALL ZEROV(MVEC2,3)
CALL ZEROV(BVEC2,3)
CALL COMPUTE(UA,UB,AKAPPA1,AKAPPA2,NVEC,AVEC,
+
ROOT1,ROOT2,MVEC1,BVEC1,MVEC2,BVEC2)
C--------------------------------------------------AKAPPA1 = -1.DO
AKAPPA2 = 1.DO
CALL ZEROV(NVEC,3)
CALL ZEROV(AVEC,3)
CALL ZEROV(MVEC1,3)
CALL ZEROV(BVEC1,3)
CALL ZEROV(MVEC2,3)
CALL ZEROV(BVEC2,3)
CALL COMPUTE(UA,UB,AKAPPA1,AKAPPA2,NVEC,AVEC,
152
ROOT1,ROOT2,MVEC1,BVEC1,MVEC2,BVEC2)
+
C-------------------------------------------------AKAPPA1 = -1.DO
AKAPPA2 = -1.DO
CALL ZEROV(NVEC,3)
CALL ZEROV(AVEC,3)
CALL ZEROV(MVEC1,3)
CALL ZEROV(BVEC1,3)
CALL ZEROV(MVEC2,3)
CALL ZEROV(BVEC2,3)
CALL ZEROV(SVEC1,3)
CALL ZEROV(SVEC2,3)
CALL COMPUTE(UA,UB,AKAPPA1,AKAPPA2,NVEC,AVEC,
+
ROOT1,ROOT2,MVEC1,BVEC1,MVEC2,BVEC2)
C--------------------------------------------------STOP
END
C---------------------------------------------------SUBROUTINE COMPUTE(UA,UB,AKAPPA1,AKAPPA2,NVEC,AVEC,
+
ROOT1,ROOT2,MVEC1,BVEC1,MVEC2,BVEC2)
IMPLICIT REAL*8(A-H,O-Z)
REAL*8 UA(3,3),UB(3,3),RAB(3,3),RH(3,3),
+
AVEC(3),NVEC(3),
+
APRIME(3),NPRIME(3),E1(3),E2(3),E3(3),
+
UAINV(3,3),C(3,3),CO(3,3),AUX1(3,3),AUX2(3,3),
+
EIGVAL(3),EIGVEC(3,3),AIDEN(3,3),NTILDE(3),
+
ATILDE(3),CLAMBDA1(3,3),CLAMBDA2(3,3),
153
BVEC1(3),MVEC1(3),BVEC2(3),MVEC2(3)
+
C
C
CONSTRUCT THE MATRIX C
C
CALL MPROD(UB,UB,AUX1)
CALL M3INV(UA,UAINV)
CALL MPROD(AUX1,UAINV,AUX2)
CALL MPROD(UAINV,AUX2,C)
C
C
CALCULATE THE EIGENVALUES AND EIGENVECTORS OF C
C
CALL SPECTRAL(C,EIGVAL,EIGVEC)
C
C
NOTE THAT SPECTRAL CALCULATES THE EIGENVALUES IN
C
DESCENDING ORDER. REORDER THE EIGENVALUES AND EIGENVECTORS
C
ALAMBDA1 = EIGVAL(3)
ALAMBDA2 = EIGVAL(2)
ALAMBDA3 = EIGVAL(1)
DO 20 I
20
=
1,3
E1(I)
=
EIGVEC(I,3)
E2(I)
=
EIGVEC(I,2)
E3(I)
=
EIGVEC(I,1)
CONTINUE
CALL ZEROV(NPRIME,3)
CALL ZEROV(APRIME,3)
154
DO 30 I=1,3
NPRIME(I) =
+
((DSQRT(ALAMBDA3)-DSQRT(ALAMBDA1))/DSQRT(ALAMBDA3-ALAMBDA1))
+
* ( - DSQRT(1.DO - ALAMBDA1)*E1(I)+
+
AKAPPA1*DSQRT(ALAMBDA3 - 1.DO)*E3(I)
30
)
CONTINUE
CALL DOTPV(NPRIME,NPRIME,RHO)
RHO = DSQRT(RHO)
RHOINV = 1.DO/RHO
DO 40 I=1,3
NPRIME(I) = RHOINV*NPRIME(I)
40
CONTINUE
DO 50 I=1,3
APRIME(I)
+
= RHO*(
DSQRT(ALAMBDA3*(1.DO-ALAMBDA1)/(ALAMBDA3-ALAMBDA1))*E1(I)+
+ AKAPPA1*
+
50
DSQRT(ALAMBDA1*(ALAMBDA3-1.DO)/(ALAMBDA3-ALAMBDA1))*E3(I))
CONTINUE
DO 60 I=1,3
AVEC(I)
60
=
APRIME(I)
CONTINUE
DO 70 I=1,3
DO 70 J=1,3
NVEC(I) = NVEC(I) + UA(I,J)*NPRIME(J)
155
70
CONTINUE
CALL DOTPV(NVEC,NVEC,AMAG)
AMAG = DSQRT(AMAG)
C
C
NORMALIZE SO THAT NVEC HAS UNIT NORM
C
DO 80 I=1,3
NVEC(I) = NVEC(I)/AMAG
AVEC(I) = AVEC(I)*AMAG
80
CONTINUE
C
C
CONTRUCT THE FUNCTION G(LAMBDA),
EQUATION 5.56 OF BALL AND JAMES
C
CALL ZEROM(AUX1)
CALL ONEM(AIDEN)
CALL MPROD(UA,UA,AUX1)
CALL MDET(AUX1,DETUA2)
DO 90 I=1,3
DO 90 J=1,3
AUX2(I,J) = AUX1(I,J)-AIDEN(I,J)
90
CONTINUE
CALL MDET(AUX2,CQUAD)
CALL ZEROV(NTILDE,3)
CALL ZEROV(ATILDE,3)
DO 100 I=1,3
DO 100 J=1,3
156
NTILDE(I) = NTILDE(I) + UAINV(I,J)*NVEC(J)
= ATILDE(I) + UAINV(I,J)*AVEC(J)
ATILDE(I)
100
CONTINUE
CALL DOTPV(NTILDE,NTILDE,AMAGNTILDE2)
CALL DOTPV(ATILDE,ATILDE,AMAGATILDE2)
CALL DOTPV(ATILDE,ATILDE,SMAG1)
CALL DOTPV(AVEC,AVEC,AMAGAVEC2)
DO 236 I=1,3
ATILDE(I) = ATILDE(I)/DSQRT(SMAG1)
236
CONTINUE
AQUAD = AMAGAVEC2 - DETUA2*AMAGATILDE2*AMAGNTILDE2
BQUAD =
-
AQUAD
DISCRIM
=
DSQRT(BQUAD*BQUAD
- 4.DO*AQUAD*CQUAD)
ROOT1
=
(-
BQUAD + DISCRIM)/(2.DO*AQUAD)
ROOT2
=
(-
BQUAD - DISCRIM)/(2.DO*AQUAD)
C
C
CONSTRUCT THE MATRICES CLAMBDA1 AND CLAMBDA2
C
CALL ZEROM(AUX1)
CALL ZEROM(AUX2)
DO 110 I=1,3
DO 110 J=1,3
AUX1(I,J) = UA(I,J) + ROOT1*NVEC(I)*AVEC(J)
AUX2(I,J) = UA(I,J) + ROOT1*AVEC(I)*NVEC(J)
110
CONTINUE
157
CALL MPROD(AUX1,AUX2,CLAMBDA1)
CALL ZEROM(AUX1)
CALL ZEROM(AUX2)
DO 120 I=1,3
DO 120 J=1,3
120
AUX1(I,J)
= UA(I,J) + ROOT2*NVEC(I)*AVEC(J)
AUX2(I,J)
= UA(I,J) + ROOT2*AVEC(I)*NVEC(J)
CONTINUE
CALL MPROD(AUX1,AUX2,CLAMBDA2)
C
C
CALCULATE THE EIGENVALUES AND EIGENVECTORS OF CLAMBDA1
C
CALL SPECTRAL(CLAMBDA1,EIGVAL,EIGVEC)
C
NOTE THAT SPECTRAL CALCULATES THE EIGENVALUES IN
C
DESCENDING ORDER. REORDER THE EIGENVALUES AND EIGENVECTORS
C
ALAMBDA1 = EIGVAL(3)
ALAMBDA2 = EIGVAL(2)
ALAMBDA3
DO 130 I
=
=
EIGVAL(1)
1,3
E1(I) = EIGVEC(I,3)
E2(I) = EIGVEC(I,2)
E3(I) = EIGVEC(I,1)
130
CONTINUE
CALL ZEROV(NPRIME,3)
158
CALL ZEROV(APRIME,3)
DO 140 I=1,3
NPRIME(I) =
+
((DSQRT(ALAMBDA3)-DSQRT(ALAMBDA1))/DSQRT(ALAMBDA3-ALAMBDA1))
+
*( - DSQRT(1.D0 - ALAMBDA1)*E1(I)+
+
AKAPPA2*DSQRT(ALAMBDA3 - 1.DO)*E3(I)
)
CONTINUE
140
CALL DOTPV(NPRIME,NPRIME,RHO)
RHO = DSQRT(RHO)
RHOINV = 1.DO/RHO
C
WRITE(20,*)'
RHO = ',RHO,'
RHOINV = ',RHOINV
DO 150 I=1,3
NPRIME(I) = RHOINV*NPRIME(I)
CONTINUE
150
DO 160 I=1,3
APRIME(I) = RHO*(
+
DSQRT(ALAMBDA3*(1.DO-ALAMBDA1)/(ALAMBDA3-ALAMBDA1))*E1(I)+
+
AKAPPA2*
+
DSQRT(ALAMBDA1*(ALAMBDA3-1.DO)/(ALAMBDA3-ALAMBDA1))*E3(I)
+
)
160 CONTINUE
DO 170 I=1,3
159
170
BVEC1 (I)
= APRIME(I)
MVEC1(I)
=
NPRIME(I)
CONTINUE
C
C
CALCULATE MVEC2 AND BVEC2 CORRESPONDING TO CLAMBDA2 AND ROOT2
C
C
CALCULATE THE EIGENVALUES AND EIGENVECTORS OF CLAMBDA1
C
CALL SPECTRAL(CLAMBDA2,EIGVAL,EIGVEC)
C
C
NOTE THAT SPECTRAL CALCULATES THE EIGENVALUES IN
C
DESCENDING ORDER. REORDER THE EIGENVALUES AND EIGENVECTORS
C
ALAMBDA1 = EIGVAL(3)
ALAMBDA2 = EIGVAL(2)
ALAMBDA3
=
EIGVAL(1)
DO 180 I
=
1,3
E1(I) = EIGVEC(I,3)
E2(I) = EIGVEC(I,2)
E3(I) = EIGVEC(I,1)
180
CONTINUE
CALL ZEROV(NPRIME,3)
CALL ZEROV(APRIME,3)
DO 190 I=1,3
NPRIME(I) =
160
+
((DSQRT(ALAMBDA3)-DSQRT(ALAMBDAl))/DSQRT(ALAMBDA3-ALAMBDA1))
+
* ( - DSQRT(1.D0 - ALAMBDA1)*E1(I)+
+
AKAPPA2*DSQRT(ALAMBDA3 - 1.DO)*E3(I)
)
CONTINUE
190
CALL DOTPV(NPRIME,NPRIME,RHO)
RHO = DSQRT(RHO)
RHOINV = 1.DO/RHO
C
WRITE(20,*)'
RHO = ',RHO,
' RHOINV = ',RHOINV
DO 200 I=1,3
NPRIME(I) = RHOINV*NPRIME(I)
200
CONTINUE
DO 210 I=1,3
APRIME(I)
= RHO*(
+
DSQRT(ALAMBDA3*(1.DO-ALAMBDA1)/(ALAMBDA3-ALAMBDA1))*E1(I)+
+
AKAPPA2*
+
DSQRT(ALAMBDA1*(ALAMBDA3-1.DO)/(ALAMBDA3-ALAMBDA1))*E3(I))
210
CONTINUE
DO 220 I=1,3
220
BVEC2(I)
= APRIME(I)
MVEC2(I)
= NPRIME(I)
CONTINUE
WRITE(21,7)BVEC1(1),BVEC1(2),BVEC1(3),
+
MVEC1(1),MVEC1(2),MVEC1(3)
161
WRITE(21,7)BVEC2(1),BVEC2(2),BVEC2(3),
+
7
MVEC2(1),MVEC2(2),MVEC2(3)
FORMAT(F7.4,4X,F7.4,4X,F7.4,4X,F7.4,4X,F7.4,4X,F7.4)
RETURN
END
C
**********************************************************************
C
C
THE FOLLOWING SUBROUTINES CALCULATE THE SPECTRAL
C
DECOMPOSITION OF A SYMMETRIC THREE BY THREE MATRIX
C
C
**********************************************************************
SUBROUTINE SPECTRAL(A,D,V)
C
THIS SUBROUTINE CALCULATES THE EIGENVALUES AND EIGENVECTORS OF
C
A SYMMETRIC 3 BY 3 MATRIX [A].
C
C
THE OUTPUT CONSISTS OF A VECTOR D CONTAINING THE THREE
C
EIGENVALUES IN ASCENDING ORDER, AND
C
A MATRIX [V] WHOSE COLUMNS CONTAIN THE CORRESPONDING
C
EIGENVECTORS.
C ---------------------------------------------------------------------IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER(NP=3)
DIMENSION D(NP),V(NP,NP)
DIMENSION A(3,3),E(NP,NP)
DO 2 I = 1,3
DO 1 J= 1,3
162
E(I,J) = A(I,J)
1
CONTINUE
2
CONTINUE
CALL JACOBI(E,3,NP,D,V,NROT)
CALL EIGSRT(D,V,3,NP)
RETURN
END
C
**********************************************************************
SUBROUTINE JACOBI(A,N,NP,D,V,NROT)
C
C
COMPUTES ALL EIGENVALUES AND EIGENVECTORS OF A REAL SYMMETRIC
C
MATRIX [A],
C
NP BY BP ARRAY.
C
ARE DESTROYED, BUT THE DIAGONAL AND SUB-DIAGONAL ARE UNCHANGED
C
AND GIVE FULL INFORMATION ABOUT THE ORIGINAL SYMMETRIC MATRIX.
C
VECTOR D RETURNS THE EIGENVALUES OF [A] IN ITS FIRST N ELEMENTS.
C
[V] IS A MATRIX WITH THE SAME LOGICAL AND PHYSICAL DIMENSIONS AS
C
[A] WHOSE COLUMNS CONTAIN, ON OUTPUT, THE NORMALIZED
C
EIGENVECTORSOF [A].
C
WHICH WERE REQUIRED.
WHICH IS OF SIZE N BY N,
STORED IN A PHYSICAL
ON OUTPUT, ELEMENTS OF [A] ABOVE THE DIAGONAL
NROT RETURNS THE NUMBER OF JACOBI ROTATIONS
C
C
THIS SUBROUTINE IS TAKEN FROM "NUMERICAL RECIPES", PAGE 346.
C ---------------------------------------------------------------------IMPLICIT REAL*8 (A-H,O-Z)
PARAMETER (NMAX =100)
DIMENSION A(NP,NP),D(NP),V(NP,NP),B(NMAX),Z(NMAX)
163
C
C
INITIALIZE [VI TO THE IDENTITY MATRIX
C
DO 12 IP = 1,N
DO 11 IQ = 1,N
V(IP,IQ) = O.DO
11
CONTINUE
V(IP,IP) = 1.DO
12
CONTINUE
C
INITIALIZE B AND D TO THE DIAGONAL OF [A],
C
THE C
C
EQUATION (11.1.14)
VECTOR Z WILL ACCUMULATE TERMS OF THE FORM T*APQ AS
DO 13 IP = 1,N
13
B(IP)
=
A(IP,IP)
D(IP)
=
B(IP)
Z(IP)
=
O.DO
CONTINUE
NROT = 0
DO 24 I = 1,99
C
C
AND Z TO ZERO.
SUM OFF-DIAGONAL ELEMENTS
C
SM = 0.DO
DO 15 IP = 1,
N-1
164
IN
DO 14 IQ = IP + 1,
N
SM = SM + DABS ( A(IP,IQ ))
14
CONTINUE
15
CONTINUE
C
C
IF SUM = 0.,
C
WHICH RELIES ON QUADRATIC CONVERGENCE TO MACHINE
C
UNDERFLOW.
THEN RETURN. THIS IS THE NORMAL RETURN
C
IF( SM .EQ. 0.DO) RETURN
C
IF( SM .LT. 1.0D-15) RETURN
C
IN THE FIRST THREE
C
SWEEPS CARRY OUT THE PQ ROTATION
C
IAPQI > TRESH, WHERE TRESH IS SOME THRESHOLD VALUE,
C
SEE EQUATION (11.1.25).
ONLY IF
THEREAFTER TRESH = 0.
IF( I .LT. 4) THEN
TRESH = 0.2DO*SM/N**2
ELSE
TRESH = 0.DO
END IF
DO 22 IP = 1, N-1
DO 21 IQ = IP+1,N
G = 100.DO*DABS(A(IP,IQ))
C
C
AFTER FOUR SWEEPS, SKIP THE ROTATION IF
C
THE
C
OFF-DIAGONAL ELEMENT IS SMALL.
165
C
IF((I .GT.
+
.AND.
4)
.AND.
( DABS(D(IP))+G
( DABS(D(IQ))+G
.EQ.
DABS(
.EQ.
DABS(
D(IQ))
) )THEN
A(IP,IQ) = 0.DO
ELSE IF( DABS(A(IP,IQ))
.GT.
TRESH) THEN
H = D(IQ) - D(IP)
IF(DABS(H)+G .EQ.
DABS(H))
THEN
C
T= 1./(2.*THETA), EQUATION (11.1.10)
C
C
T =A(IP,IQ)/H
ELSE
THETA = 0.5DO*H/A(IP,IQ)
T =1.DO/(DABS(THETA)+DSQRT(1.DO+THETA**2))
IF(THETA .LT.
0.DO) T = -T
END IF
C = 1.DO/DSQRT(1.DO + T**2)
S = T*C
TAU = S/(1.DO + C)
H = T*A(IP,IQ)
Z(IP) = Z(IP) - H
Z(IQ) = Z(IQ) + H
D(IP) = D(IP) - H
166
D(IP))
)
D(IQ) = D(IQ) + H
A(IP,IQ) = O.DO
C
CASE OF ROTATIONS 1<= J < P
C
C
DO 16 J = 1,
IP-1
G = A(J,IP)
H = A(J,IQ)
A(J,IP) = G - S*(H + G*TAU)
A(J,IQ) = H + S*(G - H*TAU)
16
CONTINUE
C
CASE OF ROTATIONS P < J < Q
C
C
DO 17 J = IP+1,
G
=
A(IP,J)
H
=
A(J,IQ)
IQ-1
A(IP,J) = G - S*(H + G*TAU)
A(J,IQ) = H + S*(G - H*TAU)
17
CONTINUE
C
CASE OF ROTATIONS Q < J <= N
C
C
DO 18 J = IQ+1, N
G = A(IP,J)
H = A(IQ,J)
18
A(IP,J)
=
G - S*(H + G*TAU)
A(IQ,J)
=
H + S*(G - H*TAU)
CONTINUE
167
DO 19 J = 1,N
G = V(J, IP)
H = V(J,IQ)
V(J,IP) = G - S*(H + G*TAU)
V(J,IQ) = H + S*(G - H*TAU)
19
CONTINUE
NROT = NROT + 1
END IF
21
CONTINUE
22
CONTINUE
C
UPDATE D WITH THE SUM OF T*APQ,
DO 23 IP = 1,
AND REINITIALIZE Z C
N
B(IP) = B(IP) + Z(IP)
D(IP) = B(IP)
Z(IP) = 0.DO
23
CONTINUE
24
CONTINUE
C
C
IF THE ALGORITHM HAS REACHED THIS STAGE, THEN
C
THERE ARE TOO MANY SWEEPS, PRINT A DIAGNOSTIC
C
AND EXIT
C
WRITE (6, '(/1X,A/)')'FROM JACOBI: 99 ITERS. SHOULD NEVER HAPPEN'
STOP
RETURN
END
C
**********************************************************************
168
SUBROUTINE EIGSRT(D,V,N,NP)
C
GIVEN THE EIGENVALUES D AND EIGENVECTORS [V] AS OUTPUT FROM
C
JACOBI, THIS ROUTINE SORTS THE EIGENVALUES INTO ASCENDING ORDER,
C
AND REARRANGES THE COLUMNS OF [VI ACCORDINGLY.
C
C
THIS SUBROUTINE IS TAKEN FROM "NUMERICAL RECIPES",
P. 348.
C ---------------------------------------------------------------------IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION D(NP),V(NP,NP)
DO 13 I = 1,N-1
K= I
P
=
D(I)
DO 11 J = I+1,N
IF(D(J)
P) THEN
.GE.
K= J
P
D(J)
=
END IF
11
CONTINUE
IF(K .NE. I) THEN
D(K)
=
D(I)
D(I)
=
P
DO 12 J = 1,N
P = V(J,I)
12
V(J,I)
=
V(J,K)
V(J,K)
=
P
CONTINUE
END IF
13
CONTINUE
169
RETURN
END
C **********************************************************************
C
C
THE FOLLOWING SUBROUTINES ARE UTILITY ROUTINES
C
C **********************************************************************
SUBROUTINE ZEROV(V,SIZE)
C
C
THIS SUBROUTINE STORES THE ZERO VECTOR IN A VECTOR V
C ----------------------------------------------------------------------
IMPLICIT REAL*8 (A-H,O-Z)
INTEGER SIZE
REAL*8 V(SIZE)
DO 1 I = 1,SIZE
V(I) = 0.DO
1
CONTINUE
RETURN
END
C **********************************************************************
SUBROUTINE ZEROM(A)
C
C
THIS SUBROUTINE SETS ALL ENTRIES OF A 3 BY 3 MATRIX TO O.DO
C ---------------------------------------------------------------------REAL*8 A(3,3)
DO 1 I=1,3
DO 1 J=1,3
170
A(I,J) = 0.DO
1
CONTINUE
RETURN
END
C
**********************************************************************
SUBROUTINE ONEM(A)
C
C
THIS SUBROUTINE STORES THE IDENTITY MATRIX IN THE
C
3 BY 3 MATRIX [A]
C ----------------------------------------------------------------------
REAL*8 A(3,3)
DO 1 I = 1,3
DO 1 J = 1,3
IF (I .EQ. J) THEN
A(I,J)
= 1.ODO
ELSE
A(I,J)
= O.ODO
END IF
1
CONTINUE
RETURN
END
C
**********************************************************************
SUBROUTINE MTRANS(A,ATRANS)
C
THIS SUBROUTINE CALCULATES THE TRANSPOSE OF AN 3 BY 3
C
MATRIX [A],
AND PLACES THE RESULT IN ATRANS.
C ---------------------------------------------------------------------171
C
VARIABLES
C
REAL*8 A(3,3), ATRANS(3,3)
C ---------------------------------------------------------------------C
COMPUTATION
C
DO 1 I = 1,
3
DO 1 J = 1,
3
ATRANS(I,J)
1
= A(J,I)
CONTINUE
RETURN
END
C **********************************************************************
SUBROUTINE MPROD(A,B,C)
C
THIS SUBROUTINE MULTIPLIES TWO 3 BY 3 MATRICES [A] AND [B],
C
AND PLACE THEIR PRODUCT IN MATRIX [C].
C ---------------------------------------------------------------------C
VARIABLES
C
REAL*8 A(3,3), B(3,3), C(3,3)
C ---------------------------------------------------------------------C
COMPUTATION
C
DO 2 I = 1,
3
DO 2 J = 1,
3
C(I,J) = 0.DO
DO 1 K = 1,
3
C(I,J) = C(I,J) + A(I,K) * B(K,J)
1
CONTINUE
172
2
CONTINUE
RETURN
END
C
**********************************************************************
SUBROUTINE MPRODVEC(D,E,F)
C
THIS SUBROUTINE MULTIPLIES A 3 BY 3 MATRIX [E] WITH VECTOR [F],
C
AND PLACE THEIR PRODUCT IN VECTOR [D].
C ---------------------------------------------------------------------C
VARIABLES
C
REAL*8 D(3), E(3,3), F(3)
C ---------------------------------------------------------------------C
COMPUTATION
C
DO 21 I = 1,
3
D(I) = 0.DO
DO 11 J = 1,
3
D(I) = D(I) + E(I,J) * F(J)
11
CONTINUE
21
CONTINUE
RETURN
END
SUBROUTINE DOTPM(A,B,C)
C
C
THIS SUBROUTINE CALCULATES THE SCALAR PRODUCT OF TWO
C
3 BY 3 MATRICES [A] AND [B] AND STORES THE RESULT IN THE
C
SCALAR C.
C ----------------------------------------------------------------------
173
C
VARIABLES
C
REAL*8 A(3,3),B(3,3),C
C= O.DO
DO 1 I = 1,3
DO 1 J = 1,3
C = C + A(I,J)*B(I,J)
1
CONTINUE
RETURN
END
C
**********************************************************************
SUBROUTINE DOTPV(A,B,C)
C
C
THIS SUBROUTINE CALCULATES THE SCALAR PRODUCT OF TWO
C
3-VECTORS [A] AND [B] AND STORES THE RESULT IN THE
C
SCALAR C.
C ----------------------------------------------------C
VARIABLES
C
REAL*8 A(3),B(3),C
C= O.DO
DO 1 I = 1,3
C = C + A(I)*B(I)
CONTINUE
RETURN
END
174
C
**********************************************************************
SUBROUTINE MDET(A,DET)
C
THIS SUBROUTINE CALCULATES THE DETERMINANT
C
OF A 3 BY 3 MATRIX [A].
C ---------------------------------------------------------------------C
VARIABLES
C
REAL*8
A(3,3),
DET
C ---------------------------------------------------------------------C
COMPUTATION
C
DET
=
A(1,1)*A(2,2)*A(3,3)
+
+ A(1,2)*A(2,3)*A(3,1)
+
+ A(1,3)*A(2,1)*A(3,2)
+
- A(3,1)*A(2,2)*A(1,3)
+
- A(3,2)*A(2,3)*A(1,1)
+
- A(3,3)*A(2,1)*A(1,2)
RETURN
END
C
**********************************************************************
SUBROUTINE M3INV(A,AINV)
C
C
THIS SUBROUTINE CALCULATES THE THE INVERSE OF A 3 BY 3 MATRIX
[A] AND PLACES THE RESULT IN [AINV].
C
IF DET(A) IS ZERO, THE CALCULATION
C
IS TERMINATED AND A DIAGNOSTIC STATEMENT IS PRINTED.
C ---------------------------------------------------------------------C
VARIABLES
175
C
REAL*8
AINV(3,3),
A(3,3),
--
DET, ACOFAC(3,3),
THE MATRIX WHOSE INVERSE IS DESIRED.
C
A(3,3)
C
DET --
C
ACOFAC(3,3)
C
THE SIGNED MINOR (-1)**(I+J)*M\{IJ\}
C
IS CALLED THE COFACTOR OF A(I,J).
C
AADJ(3,3)
C
OBTAINED BY REPLACING EACH ELEMENT OF
C
[A] BY ITS COFACTOR, AND THEN TAKING
C
TRANSPOSE OF THE RESULTING MATRIX.
C
AINV(3,3)
C
[AINVI = [AADJ]/DET.
THE COMPUTED DETERMINANT OF [A].
--
--
--
THE MATRIX OF COFACTORS OF A(I,J).
THE ADJOINT OF [A].
IF ( DET .EQ.
IT IS THE MATRIX
RETURNED AS INVERSE OF [A].
C
CALL MDET(A,DET)
0.DO ) THEN
WRITE(6,10)
STOP
END IF
CALL MCOFAC(A,ACOFAC)
CALL MTRANS(ACOFAC,AADJ)
DO 1 I = 1,3
DO 1 J = 1,3
AINV(I,J) = AADJ(I,J)/DET
1
AADJ(3,3)
CONTINUE
176
C ---------------------------------------------------------------------C
FORMAT
C
10 FORMAT(5X, '--ERROR IN M3INV--- THE MATRIX IS SINGULAR',!,
+
1oX, 'PROGRAM TERMINATED')
C--------------------------------------------------------------------RETURN
END
C
SUBROUTINE MCOFAC(A,ACOFAC)
C
THIS SUBROUTINE CALCULATES THE COFACTOR OF A 3 BY 3 MATRIX [A],
C
AND PLACES THE RESULT IN ACOFAC.
C ---------------------------------------------------------------------C
VARIABLES
C
REAL*8 A(3,3), ACOFAC(3,3)
C ---------------------------------------------------------------------C
COMPUTATION
C
ACOFAC(1,1) = A(2,2)*A(3,3)
ACOFAC(1,2) = -(A(2,1)*A(3,3)
ACOFAC(1,3) = A(2,1)*A(3,2)
ACOFAC(2,1)
= -(A(1,2)*A(3,3)
ACOFAC(2,2)
= A(1,1)*A(3,3)
ACOFAC(2,3) = -(A(1,1)*A(3,2)
ACOFAC(3,1) = A(1,2)*A(2,3)
ACOFAC(3,2) = -(A(1,1)*A(2,3)
- A(3,2)*A(2,3)
- A(3,1)*A(2,3))
- A(3,1)*A(2,2)
- A(3,2)*A(1,3))
- A(3,1)*A(1,3)
- A(3,1)*A(1,2))
- A(2,2)*A(1,3)
- A(2,1)*A(1,3))
177
ACOFAC(3,3)
= A(1,1)*A(2,2)
- A(2,1)*A(1,2)
RETURN
END
C
**********************************************************************
SUBROUTINE INVAR(A,IA,IIA,IIIA)
C
C
THIS
C
IA,
SUBROUTINE CALCULATES THE PRINCIPAL INVARIANTS
IIA, IIIA OF
A
TENSOR [A].
C ----------------------------------------------------------------------
C
VARIABLES
C
REAL*8 A(3,3),
AD(3,3),AD2(3,3),
DETA, IA,IIA,IIIA
C
C
COMPUTATION
C
DO 1 I=1,3
DO 1 J=1,3
AD(I,J) = A(I,J)
1
CONTINUE
IA = AD(1,1) + AD(2,2) + AD(3,3)
C
C
CALCULATE THE SQUARE OF [AD]
C
CALL MPROD(AD,ADAD2)
C
IIA =0.5DO * ( IA*IA - ( AD2(1,1)
C
178
+ AD2(2,2)
+ AD2(3,3)
) )
CALL
MDET(AD,DETA)
IIIA = DETA
RETURN
END
C
**********************************************************************
SUBROUTINE TRACEM(A,TRA)
C
C
THIS SUBROUTINE CALCULATES THE TRACE OF A 3 BY 3 MATRIX [A]
C
AND STORES THE RESULT IN THE SCALAR TRA
C ---------------------------------------C
VARIABLES
C
REAL*8 A(3,3),TRA
TRA = A(1,1)
+ A(2,2)
+ A(3,3)
RETURN
END
C
**********************************************************************
SUBROUTINE DEVM(A,ADEV)
C
C
THIS SUBROUTINE CALCULATES THE DEVIATORIC PART OF A
C
3 BY 3 MATRIX [A]
C ---------------------------------------------------C
VARIABLES
C
REAL*8 A(3,3),TRA,ADEV(3,3),IDEN(3,3)
179
CALL TRACEM(A,TRA)
CALL ONEM(IDEN)
CALL ZEROM(ADEV)
DO 1 I = 1,3
DO 1 J = 1,3
ADEV(I,J)
1
=
A(I,J) - (1.DO/3.DO)*TRA*IDEN(I,J)
CONTINUE
RETURN
END
C
SUBROUTINE EQUIVS(S,SB)
C
C
THIS SUBROUTINE CALCULATES THE EQUIVALENT TENSILE STRESS SB
C
CORESSPONDING TO A 3 BY 3 STRESS MATRIX [S]
C ------------------------------------------C
VARIABLES
C
REAL*8 S(3,3),SDEV(3,3),SDOTS,SB
SB = O.DO
SDOTS = 0.DO
CALL DEVM(S,SDEV)
CALL DOTPM(SDEV,SDEV,SDOTS)
SB = DSQRT(1.5DO* SDOTS)
RETURN
180
END
SUBROUTINE PRTMAT(A,M,N)
INTEGER M,N
REAL*8 A(M,N)
DO 10 K=1,M
WRITE (23, '(2X,6F12.6,2X) ') (A(K,L), L=1,N)
10
CONTINUE
RETURN
END
SUBROUTINE PRTVEC(A,M)
INTEGER M
REAL*8 A(M)
WRITE (20, '(2X,6F12.6,2X)' ) (A(K),
K=1,M)
RETURN
END
SUBROUTINE SKINEM(F,R,U,ITERERR)
C
C
THIS SUBROUTINE PERFORMS THE RIGHT POLAR DECOMPOSITION
181
C
[F]=[R] [U] OF THE DEFORMATION GRADIENT [F] INTO
C
A ROTATION [R] AND THE RIGHT
C
IS ALSO CALCULATED.
STRETCH TENSOR [U].
C ---------------------------------------------------C
VARIABLES
C
IMPLICIT REAL*8 (A-H,O-Z)
REAL*8 F(3,3),DETF,FTRANS(3,3),
+
C(3,3),
+
U(3,3),UINV(3,3),R(3,3),TEMP(3,3)
OMEGA(3),EIGVEC(3,3),
EIGVECT(3,3),
C -------------
C
COMPUTATION
C
C
C
STORE THE IDENTITY MATRIX IN [R],
[U],
AND [UINV]
C
CALL ONEM(R)
CALL ONEM(U)
CALL ONEM(UINV)
C
C
STORE THE ZERO MATRIX IN [E]
C
CALL ZEROM(E)
C
C
CHECK IF THE DETERMINANT OF [F] IS GREATER THAN ZERO.
C
IF NOT, THEN PRINT DIAGONOSTIC AND EXIT.
C
182
CALL MDET(F,DETF)
IF (DETF .LE.
0.DO) THEN
ITERERR = 1
RETURN
END IF
C
C
CALCULATE THE RIGHT CAUCHY GREEN STRAIN TENSOR [C]
C
CALL
MTRANS(F,FTRANS)
CALL
MPROD (FTRANS, F,C)
C
C
CALCULATE THE EIGENVALUES AND EIGENVECTORS OF
C
CALL SPECTRAL(C,OMEGA,EIGVEC)
C
C
CALCULATE THE PRINCIPAL VALUES OF [U] AND [E]
C
U(1,1)
= DSQRT(OMEGA(1))
U(2,2)
= DSQRT(OMEGA(2))
U(3,3)
= DSQRT(OMEGA(3))
C
C
CALCULATE THE COMPLETE TENSORS [U] AND [El
C
183
[C]
CALL MTRANS(EIGVEC,EIGVECT)
CALL MPROD(EIGVEC,U,TEMP)
CALL MPROD(TEMP,EIGVECT,U)
C
C
CALCULATE [UINV]
C
CALL
C
M3INV(U,UINV)
CALCULATE [R]
CALL MPROD(F,UINV,R)
C ---------------------------------------------------------------------C
FORMATS
C ---------------------------------------------------------------------RETURN
END
184
Appendix E
Time-integration procedure :
Isotropic-based constitutive model
In this appendix we summarize the time-integration procedure that we have used for
our rate-independent isotropic-based constitutive model. With t denoting the current
time, At is an infinitesimal time increment, and T
Given: (1) {F(t), F(T),
=
t+At, the algorithm is as follows:
(2) {T(t), FP(t)}; (3) the accumulated martensite
0(t), O(r)};
volume fractions (t).
Calculate: (a){T(r), FP(T)}, (b) the accumulated martensite volume fractions (T),
and (c) the inelastic work increment Aw
(T)
and march forward in time.
The steps used in the calculation procedure are:
Step 1. Calculate the trial elastic strain
Fe (T)trial
ce (T)trial =
Ee(T)trial
-
=
Ee()trial:
F(T)(FP(t))
(Fe (T)trial )TFe
(r)trial,
(1/2) {Ce(T)trial
1}.
Step 2. Calculate the trial stress Te(T)trial:
Te(T)trial = C(t)[Ee(T)trial -
185
Cf(t)(O(T) - 00)1].
Step 3. Calculate the phase transformation flow direction NP and the trial resolved
force
&(T)trial :
NP =
(3/2)T
NP = V/(3/2)
if
e
IET (IetcI
if
(t) = 0,
0 <
(t) < 1.
0T(t)
The trial resolved force is given by
&(,
Te ()trial
trial
-
NP.
Step 4. Calculate the trial driving force for phase transformation
f(T)frial
=
a(T)triaI
-
(AT/OT)(O(T)
-
f(r)trial:
OT).
Step 5. Calculate
(E.1)
FP(T) = {1 + A NP} FP(t).
During phase transformation, the consistency conditions must be satisfied :
(E.2)
f(7) - f, = 0,
where the - sign holds during forward transformation and the + sign holds during
reverse transformation, and where
f (T) = &(7) -
(AT/OT)(0(T)
-
(E.3)
OT).
Retaining the terms of the first order in A , it is straightforward to show that
&(,)
=
()tria
- 2p(t)A sym(Ce(T) rialNP) . NP
(E.4)
Using (E.3) and (E.4) in the consistency conditions (E.2) we can calculate AZ.
Step 6. Determine the increment of the martensite volume fraction AL
If
f(T)trial -
f,
> 0
and 0 <
:
(t) < 1 then for the forward austenite to martensite
transformation
f(r)trial
-
fc
2p(t) sym(Ce(T)trialNP) - NP'
186
If
f(,tri"
+ fc
<
0, and 0 <
(t) < 1 then for the reverse martensite to austenite
transformation
f(T) t rial + fc
2p(t) sym(Ce (T)trialNP) - NP
Or else there is no phase transformation
A\ = 0.
Step 7. Update the inelastic deformation gradient FP(r):
FP(r) = {1 + A NP} FP(t).
Step 8. Update the total martensite volume fraction
(t) + Az.
(T) =
If (r) > 1, then set
Step 9.
(T) = 1 and if
(r):
(T) < 0, then set
(T) = 0.
Update the effective elastic modulus C(T) and the coefficient of thermal
expansion a(T):
C(T) = {1
-
(T)} Ca +
{1
-
()}
a(T)
=
aa
+
(r)Cm
(),m.
Step 10. Compute the elastic deformation gradient Fe(T) and the stress Te(r):
Fe (T)
Fe(r)
Ce(T)
(r)(FP(T))-',
=
=
=
Re(T)Ue(T),
(Fe(r)) T Fe (T),
E e (T) = (1/2) {Ce (T) - 1},
Te(T)
=
C(-)[Ee(,) - A(T)(6(T) - 00)].
Step 11. Update the Cauchy stress T(T):
187
T(r)
Step 12.
=
Re(T)Te(T)Re(T)
'
.
Calculate the driving force for phase transformation
work fraction AwP(T):
d(T) =
f(r)
=
&(T) -
AwP(T) = {f
Te(F) - NP,
(AT/OT)(O(r) -
OT),
(r) + (AT/OT) 6(-r)} A.
188
f(T)
and inelastic
Appendix F
Hypo-elastic form for the
isotropic-based constitutive
equations
In this appendix the isotropic-plasticity-based constitutive equations for superelastic shape-memory materials is formulated in the hypo-elastic rate-form.
This is
done to facilitate the derivation of the Jacobian matrices to be implemented in
the ABAQUS/Standard (1999,2001) finite-element package. The solution of the two
methods (hyper-elastic vs. hypo-elastic) is identical because the elastic strains experienced in these materials are very small.
The advantage of using an implicit finite-element formulation (ABAQUS/Standard)
is that large time steps can be taken while maintaining quasi-static loading conditions. For example the Ti-Ni bio-medical stent calculation performed in Chapter
2 took just 2 hours to complete using ABAQUS/Standard (1999,2001). Performing
the same calculation using the ABAQUS/Explicit (1999,2001) finite-element program
took 25 hours. This is because large time steps cannot be used while maintaining
quasi-static loading conditions.
With that, the governing variables in the hypo-elastic constitutive model are taken
to be
189
T
L
Cauchy stress
=
D+W
Velocity gradient
Absolute temperature
0
Inelastic stretching
Martensite volume fraction
Elastic stretching
De - D - DP
TV =t-WT+TW Jaumann derivative of T
Constitutive equation for stress
TV = C [D - AO - DPI
C = 2pI+ (i, - (2/3)L) 1 0 1,
A = al,
K
a =&( , 0)
=
k( , 0)
Elasticity tensor
Thermal expansion tensor
Vol. Frac of Martensite
Transformation criteria
Let
f
denote the driving force for transformation, and
fc > 0
190
a critical value for the driving force. Then the transformation criteria are taken as
f = f,
for austenite to martensite,
> 0,
f
for martensite to austenite,
< 0.
and
=
-fc
The driving force for transformation is defined by
f = F-
(AT/OT)(O - OT),
where
Resolved tensile force
AT
Latent heat per unit volume
ET
Transformation strain
OT
Phase equilibrium temperature.
The resolved tensile force is defined by:
& = T - NP.
Here NP is the transformation direction after the transformation criterion
f = f, for
forward transformation is first satisfied and there is a non-zero . That is, the stress
used to evaluate the transformation direction NP is the stress at the time t, at which
the forward transformation criterion is first satisfied. For a given forward-reverse
transformation the direction NP is fixed.
Flow rule
DP=
The transformation direction NP is
191
NP.
eT (To)
NP=
NP =
if
()
e
if
=,
0 <( <1.
Here, t, is the critical instant when the martensite volume fraction first becomes
non-zero.
For forward transformation:
(f - fC)
0,
and
(f-
fc)=O
(f + fc) ;> 0,
and
(f+
fc)=0
For reverse transformation:
< 0,
Consistency conditions
During forward transformations
~(
= 0 if
-fc)
(f -
fc) = 0
During reverse transformations :
(f + fc) = 0
if
(f + fc) = 0
The consistency conditions serve to determine the transformation rates d.
For transformation
(F.1)
0 = (f ± fc)
(F.2)
=cr- (AT/T)( - OT) ± fc)
(F.3)
= a - (AT/OT)O
(F.4)
(AT/OT) 0
(F.5)
=''-NP - (AT/OT)O
(F.6)
=TV - NP - (AT/OT) 0
(F.7)
=
=
T NP -
C
[D
-
A
- DP] - NP -
192
(AT/OT)
0
(F.8)
Let
T* = C [D - A]
denote a trial stress rate,
=C D - AI - NP = C [D] -NP = C [NP] -D,
or
2NP - D,
=
and a trial resolved force, and
(AT/O)
b*-
=
the rate of change of a trial driving force. Then
0 =
-
C [DP]
NP
(F.9)
0 =
-
C [NP]
DP
(F.10)
0 =
- 2
N-
P
(F.11)
0 =
-2NP
( NP)
(F.13)
/i- e T7,
0 =
or
(
3 1*
CT)
3y
193
(F.12)
T
Next
TV =C D - A
(F.14)
- DP
(F.15)
= C [D - DP] - C [A]O
=C[D=C
(DNP)
(
D
=C [D] = C [D]-
C [D]
[A]
(F.17)
2pNP) -O C [A]
(F.18)
NP)
-C
( )
3p
(f
2T
NP)
= C [D] - ((2/3)
=
(F.16)
-6C [A]
{*
- (4p/3E)
(F.19)
C[A]
)T)
(
}
(F.20)
NP) - aC [A]
(NP -D) NP) + {(2/3)
(AT/
TOT)
NP - C [A]}
(F.21)
or
TV =,C [D] +{(2/3) (AT/e TOT) NP - C [A]}
where
c = C - (4p/3eT)N ®NP.
F.1
Incremental form
Let
Ft(T)= F(T)(F(t))-1
(F.22)
Ft(T) = Rt(T)Ut(r)
(F.23)
Et(r)
=
ln Ut(w)
T(T)
=
(Rt
())
(F.24)
T
194
(T(T))(Ret(r))
(F.25)
Constitutive equation for stress
T(T)
T(t) + C(t) [Et(T) - A(t) (0(T)
=
-
C(t) = 2p(t)T + (K(t) - (2/3)p(t)) 1 0 1
A(t)
=
j(((t), 0(
A(t) = a(t)l,
E'(7) = A NP,
Elasticity tensor
Elastic moduli
K ) = k((), 0(t))
Thermal expansion tensor
a(t) = &( (t), 0(t))
NP
= V/(3/2)
'ET
0(t)) - E'(T)]
''o (tc)
Inelastic strain increment,
To(tc) I
where the stress used to evaluate the transformation direction NP is the stress at the
time t, at which the forward transformation criterion
f*(r)
>
f, is first
below.
Trial driving force for phase transformation
Trial value of stress:
T*(T)
=
T(t) + C(t) [Et(T) - A(t) (0(T)
-
0(t))]
Trial value of driving force:
f*(T)
=
&*(T) -
(AT/OT)(O -
OT),
where
Trial resolved force
Transformation strain
AT
Latent heat per unit volume
Phase equilibrium temperature
195
satisfied; see
The trial resolved force is defined by:
&* (T)= T*(r)
- NP
Here NP is the transformation direction after the transformation criterion f*(r) > fc
t. That is, the stress used
for forward transformation is first satisfied at time t
to evaluate the transformation direction NP is the stress at the time tc at which the
forward transformation criterion is first satisfied, and
becomes positive valued. For
a given forward-reverse transformation the direction NP is fixed. The transformation
direction NP is
NP=
'E
!
=0,
0(t)
if
Jio(t)
2
NP
T
To(t)
if 0<
ITo(tc)I
(t)<1.
Transformation criteria
If 0 < (t) < 1 then, for austenite to martensite, j > 0:
f*(r) > fc.
If 0 < (t) < 1 then, for martensite to austenite, j < 0:
f*(T) <
-fc.
Consistency conditions
f(7)
=
5(-)
-
(AT/OT)(0
196
-
OT)
&(-F)
= T(-F) - NP
Resolved force,
Latent heat per unit volume,
AT
Transformation strain,
Phase equilibrium temperature.
During forward transformations :
f(T)
= fc
During reverse transformations :
The consistency conditions serve to determine the transformation increments A .
F.2
Time integration procedure
Here we suppress the argument t in p(t) until the end.
Ef (r) = A
T(T)
=
o(t,)
|To (tc)|
NP = V/(3/2) CT
NP,
T(t) + C(t) [Et(r) - A(t)(0(r) - 6(t))] - A C(t) [NP]
T(T) = T(t) + C(t) [Et(T) - A(t)(0(r) - 0(t))] - 2p A NP
T*(T)
=
T(t) + C(t) [Et(T) - A(t)(6(T)
T(T)
=
T*(T)
-
2p A NP
&*(r) -3PA
6(t))]
(F.27)
(F.28)
(F.29)
(F.30)
T(T) - NP = T*(T) -NP - 2pANP- NP
F(T) =
-
(F.26)
(F.31)
(F.32)
E2
197
Consistency condition for forward transformation:
f (T)
(T) -
3pA E' - (AT/OT) (0(T)
(-)
&*(T) -
-
(AT/ET) (0(F) (AT/6T) (0(T) -
=
-
OT) =
T)
-- fc =
0T)} -
f,
(F.33)
fc
(F.34)
3pA e
(F.35)
fc =
(F.36)
3p 6
(*eT)
=
-
f
3,a(t) ET
(F.37)
where
f*(T)=
d*(r) - (AT/OT)(0(T)
-
OT)
is trial driving force for phase transformation.
Consistency condition for reverse transformation:
(F.38)
f (T) =-f
(-)
-
3pA
1
E2 - (AT/OT) (O(T) - OT)
(T)-
(AT/ET) (0(T) - OT) + fc
{d*(T) -
(AT/OT) (0(T) -
OT)} + fc
-=
fc
(3p) A E2e
=
3p E 2
I\ ZecT) f*(T)fc
3/,a(t) ET
(AET
198
(F.39)
(F.40)
(F.41)
(F.42)
F.3
Jacobian Matrices
For a coupled temperature-displacement analysis four Jacobian matrices need to be
supplied into the numerical algorithm. They are
0T(r)
M(t)
.Af(t)
O(t)
T(T)
aAO
=
=
ST(r)
a0(T)
a(RPL(T))
aAE
0(RPL(r))
OAO
_(RPL(r))
06(T)
where RPL(r) is the rate of volumetric heat generation at end of the step. It is given
by
RPL(T) = {f(r) +
(AT/OT)
0(r)} (Ac/At).
(F.43)
Here we suppress the argument t in p(t) until the end.
Derivation of L(t)
T*(T) =
T(t) + C(t) [Et(-r) - A(t)(0(T)
NP = V/(3/2)T (To(tc)
*(T)
=
0(t))]
(F.44)
(F.45)
(F.46)
T*(r) - NP
T(T) = T*(T) - 2pA
-
NP
Let
AE = Et(7),
199
(F.47)
then the Jacobian is
A)(T)
(F.48)
DAB
_
/f(AF(NP)
D*(r)
DAE
2p
(F.49)
A
aAE
C(t) - 21L
(F.50)
.
Recall that
fT=3 )
A
fC
f*(T) + fC
A~3TET
for forward transformation
(F.51)
for reverse transformation.
(F.52)
Thus
D(A ET)
DAE
_ 1
a(A ET)
DAE
a(A ET)
DAE
1
3
a(A IET)
DAE
3
Df* (T)
(F.53)
DAE
-
(AT/OT)
(6(T)
-
DAE
Da* (T)
perT DAE
1
OT)
)
(F.54)
(F.55)
1
D(T*(T) - NP)
pET
DAE
(F.56)
Note that for a function f(T(E))
Df
(T(E + aA))} Ia=o
d
OT
Of
f
(T
DB
Of
(T O
d {T(E + aA)} la=o
DT
O [A]
T [
)T[
f
DT
)[9T
200
1A
].A
(F.57)
(F.58)
(F.59)
(F.60)
(F.61)
Thus
T(i*(T) - NP)
aAE ) T
aAE
=
~[ I
o
aT*(T)
(T*( T ) - NP)
OT*(T)
(F.62)
(F.63)
C(t) [NP]
(F.64)
= 2A NP.
Hence
0( ~CT)
1
aAE
3 pT r
a(A
T)
- NP)
&(*(T)
(F.65)
AE
= (2/3T) NP,
(F.66)
and the Jacobian is
C(t) = C(t) =
(2/CET) [NP 0
O(A CT)
aA2E
I
C(t) - (2pl/6') [NP 0 (2/3) NP]
(F.67)
(F.68)
or finally
L(t) = C(t) - (4pL(t)/3
2)
NP
(F.69)
ON.
Derivation of A4(t)
From equation F.47 the Jacobian matrix A4(t)
M(t) =
(F.70)
a(A NP)
DT*(2)
09(T)
=
2
(F.71)
6(T)
-C(t)[A(t)] - 2M [(1/ET)
a L\
ET)
NP].
(F.72)
From equations F.51 and F.52
M(t)
=
-C(t)[A(t)] - ( 2 ft) (1/3p 6)
06(T)
NP].
(F.73)
Since
*/(T)
-=
201
-f
-A/T
(F.74)
We obtain
M(t)
=
-3K(t)a(t)1 + (2/3)
(AT/OT6T)
(F.75)
NP.
Derivation of Af(t)
From equation F.43 the Jacobian matrix Af(t)
Af(t)
(F.76)
(RPL(r))
&9AE
=
-
fT)(A
.AT (F.77)
/At) + {f(r) + (AT/OT) O(T)} (1/ETAt)
From the consistency conditions, f(T) = Ff' - constant. Therefore
f(T)
= 0.
(F.78)
Using equations F.66 and F.78 yield :
Af(t)
=
(2/3) (1/6'At) {f(r) + (AT/OT)
9(r)}
NP.
(F.79)
Derivation of O(t)
Again from equation F.43 the Jacobian matrix O(t):
O(t)
=
(F.80)
9(RPL(T))
aO(-r)
9{Of()
(A /,At) + {f (T) + (AT/OT)
+ (AT/OT)
(T)}(1/ETLAt)
(F.81)
From the consistency conditions,
-FfC = constant. Therefore
f(T)=
f
=0
=(T)
(F.82)
99(T)
From equations F.51, F.52 and F.74 we obtain
aA
T
-
-(1/3pET)
(AT/
6
T).
(F.83)
Therefore using equations F.82 and F.83 the Jacobian matrix O(t) is
O(t) = (AT/OTAt) [A
-
(1/3p(t)E2) {f (T) + (AT/OT) 0(r)}]
202
.
(F.84)
Appendix G
Simple shear problem in the
isotropic-based constitutive model
Consider the simple shear of a single element in the (1, 3)-plane, Figure G-1a. Velocity
boundary conditions are applied to the top nodes to result in a shear strain rate of
+0.001/sec for 110 sec, and reverse deformation is prescribed to take place at a
shear strain rate of -0.001/sec in another 110 sec. The material parameters for the
constitutive model are taken to be the same as in Section 3, but this time a standard
Mises-type flow rule is used in which the direction of plastic flow is taken to be given
by
NP = V/(3/2)
Te
IT
0
for all
0 <
< 1,
(G.1)
instead of (3.15). The solution for the non-zero components of the Cauchy stress T as
a function of time is shown in Figure G-1b. Note that in addition to the shear stress
T13, non-zero and substantially large values of normal stresses T33 and T11 develop
during the reverse deformation.
This stress history results in a shear stress versus shear strain response shown
in Figure G-1b, which does not have the character of a closed flag-type superelastic
response. Figure G-1c also shows the result of a calculation for the same problem,
but upon using the non-standard flow rule (3.15) used in the body of the paper; this
time one obtains the expected closed hysteresis loop.
203
The spurious result using the standard flow rule (G. 1) occurs due to subtle normalstress effects that arise upon using a large deformation theory in simple shear. The
non-standard flow rule (3.15) predicts full recovery of stress upon complete reversal
of deformation, at least for the proportionalloading conditions tested in this section.
1
'A non-standard flow rule with a stress-based flow direction for forward transformation, and
a strain-based flow direction for reverse transformation has been previously used by Qidwai and
Lagoudas (2000), but they do not provide a discussion or a motivation for why they made their
particular choice.
204
-
I
-
I
-
-
I
I
1
-
I
I
3
j
2
_1
(a)
500
400
300|
200
100
0
F-
-100
T13
- - T33
-200
T11
-300
-400
50
0
100
150
200
TIME [s]
(b)
APnr
400
-
350
-
-
FLOW RULE
MISES-TYPE FLOW RULE
U)
Cl) 300
I- 250
C20
<200
-/
U)
-/
IOU
-/
100
-
50
__
-
(U
0
0
0.02
0.04
0.06
0.08
SHEAR STRAIN
0.1
0.12
(c)
Figure G-1: Comparison of the shear-stress versus shear-strain response in simple
shear predicted by the non-standard flow rule versus that predicted by the standardMises type flow rule. The nonstandard flow rule predicts the expected closed flag-type
hysteresis loop.
205
Appendix H
Anisotropic superelasticity of
textured sheet Ti-Ni
Here we shall evaluate our recently developed crystal-mechanics-based constitutive
model to examine the in-plane anisotropic response shown by another polycrystalline
Ti-Ni sheet. Shan and Sung (2000) have conducted extensive superelastic tension
experiments on Ti-Ni sheets along different directions, and the experimental results
show the in-plane anisotropic superelastic response. The purpose of this section is
to show that our developed crystal-mechanics-based constitutive model adequately
captures the in-plane anisotropy for a different Ti-Ni sheet.
Superelastic Ti-Ni sheets were obtained from a commercial source. Tensile specimens were cut along different directions and tested, ranging from 0' (rolling direction)
to 900 (transverse direction) at 100 intervals. Superelastic tension experiments were
conducted at room temperature (6
=
298K) under displacement control at very low
strain rates to ensure near isothermal testing conditions.
Experimental pole figure measurements of the initially-textured sheet were obtained using a Rigaku 200 X-ray diffraction machine. The {111}, {110} and {100}
experimental pole figure is shown in Figure H-la. A numerical representation of the
experimental pole figures using 420 discrete unweighted crystal orientations was obtained by using the computer program PoPLa (Kallend et al., 1989). The numerical
206
representation of the experimental pole figures is shown in Figure H-1b.
The initial finite-element mesh used to model a representative volume element
(RVE) of the polycrystalline material is shown in Figure H-2a. In our finite-element
model of a polycrystal, each element represents a single crystal, and it is assigned a
crystal orientation from the set of crystal orientations which approximate the initial
crystallographic texture of the material shown in Figure H-la.
The material parameters in the constitutive model were calibrated to the tension
experiment conducted along the rolling direction. The procedure to determine the
material parameters is outlined in Section 2.2.
The thermo-elastic constants used in our calculations are
Elastic moduli for austenite: Ca = 130 GPa, C"2
Elastic moduli for martensite: C'
=
Coefficients of thermal expansion: aa
=
98 GPa, CL4= 22 GPa;
65 GPa, C'") = 49 GPa, Cm
=
11
x
=
11 GPa;
10-6/K, a m = 6.6 x 10-6/K.
An estimate for the phase transformation parameters {OT,
AT,
fc}
for Ti-Ni single
crystals is obtained by fitting the constitutive model to tension experiment conducted
along the rolling direction. Since the experiments were only done at one particular temperature we can obtain the temperature dependence of the driving force
(AT/OT)(0 -
GT)
for the aforementioned temperature.
The single-crystal material parameters used to obtain this fit are:
Temperature dependence of driving force : (AT/Ocr) (0
-
OT) = 13.3 MJ/m 3 at
0
=
298 K.
Critical driving force: f, = 6.9 MJ/m 3 .
The fit of the constitutive model to the tension experiment conducted along the rolling
direction is shown in Figure H-2b. With the constitutive model calibrated to the
rolling direction, the superelastic tensile response along other directions (Figures H-3
to H-7) can be predicted. The superelastic response in all these orientations is wellapproximated by the predictions from the constitutive model. Of particular note,
the anisotropic response between the rolling and transverse direction, as shown in
Figure H-7b, is captured by the model. In Figure H-8 we plot the nucleation stresses
during forward and reverse transformation for the differently-oriented specimens at a
207
representative strain of 2%. Note that the nucleation stresses in the middle-oriented
specimens i.e. the 400 and 50' specimen lies in between of the rolling and transversedirection specimens. This effect is also predicted reasonably well by the constitutive
model.
To the best of our knowledge, this is the first time such calculations have been
reported in the literature.
208
3.00
2.58
2.16
1.74
1.32
.90
.48
.06
I
TD
ff
{11}
{111}
Ow"SRD
1110}1
{110}
{100}
{100}
(b)
(a)
Figure H-1: (a) Experimentally-measured texture in the as-received Ti-Ni sheet, and
(b) its numerical representation using 420 discrete unweighted crystal orientations.
209
(a)
700
600-
o EXPERIMENT:ROLLING
-
FIT
500c
400Cl)
300-
200-
100 -
0
00
0.07
0.06
0.05
0.04
0.03
STRAIN
(b)
Figure H-2: (a) Undeformed mesh of 420 ABAQUS C3D8R elements used to represent
a textured polycrystal aggregate. (b) Superelastic stress-strain curve in tension along
the rolling direction. The experimental data from this test were used to estimate
the constitutive parameters. The curve fit using the full finite-element model of the
polycrystal is also shown.
0
0.01
0.02
210
700
.I
I
6001[
o
-
,I
,I
,
EXPERIMENT : 100
PREDICTION
500.
0z
C/)
C/)
C,)
400.
-0
00
300.
~/
9
2001 -d
/
***0000
100
0I
0
0.04
0.03
STRAIN
0.02
0.01
0.06
0.05
0.07
(a)
700
600
o EXPERIMENT :20
-
-
PREDICTION
500
00
0 0 0 a 0 0 00c00c00
0 0C
W.
<) 400
UC)
w
6<0
CE
S300
d
200
-
-
100
0.01
00
0.04
STRAIN
(b)
0.05
0.06
0.07
Figure H-3: (a) Superelastic experiment conducted along the (a) 10' and (b) 200
direction. The prediction from the finite-element simulations are also shown.
211
700
600
o EXPERIMENT :300
-
-
500
0o
CL)
Cl)
PREDICTION
-
400
0
w
300
0
d
200 -q
9
100
0
0.01
0
0.04
0.03
STRAIN
(a)
0.02
0.05
0.06
0.07
0.06
0.07
70C
60C
o EXPERIMENT :40
-
ca
C,)
-
PREDICTION
50C
40C
-
00
0ee
0
0
w
0
0
C/) 30C
20C
d
q
0
10(C 0,9 ~aao o0C-0
0-000Z6 6-0
o9
0l
0
0.01
0.02
0.04
0.03
0.05
STRAIN
(b)
Figure H-4: (a) Superelastic experiment conducted along the (a) 300 and (b) 400
direction. The prediction from the finite-element simulations are also shown.
212
700
600
o EXPERIMENT :50
-
0___
-
500
PREDICTION
0o
0)
400
ooooo0ooooocc/,
0
w)
0/
I-
300
9
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q0
200
d-a
00go*0-.
100
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0. 0-O-0~
-gP
0
0
0.01
0.02
0.04
0.03
0.05
0.06
0.07
0.06
0.07
STRAIN
(a)
700
600
o
-
-
500
r-"
W
400
EXPERIMENT :600
PREDICTION
0p0/~p~%Z~
00
-9 of
300
0'
200
-
9
100 19 00-
0 --
4-0 -G
I.
0 G-oc
0
0
0.01
0.02
0.03
0.04
0.05
STRAIN
(b)
Figure H-5: (a) Superelastic experiment conducted along the (a) 500 and (b) 60'
direction. The prediction from the finite-element simulations are also shown.
213
-1
700
I
I
I
0.05
0.06
600
o EXPERIMENT :70
- PREDICTION
500
U, 400
C3)
t5
0
o/
01
300
0/
0
0
00
0/
-
200
ql
-
100
0
0.01
0
0.02
0.04
0.03
0.07
STRAIN
(a)
700
600 1o EXPERIMENT :800
-
500
-
PREDICTION
co
400
w
C,)
09
0/
-
-
300
200
d
/
%oo0000000
-U---
100
0
- -
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
STRAIN
(b)
Figure H-6: (a) Superelastic experiment conducted along the (a) 70' and (b) 800
direction. The prediction from the finite-element simulations are also shown.
214
700
600
o
-
-
EXPERIMENT : TRANSVERSE
PREDICTION
5001
C,)
400
CD)
--
Cc)
300
--d-- o?
200
1U 0
0
0
0.01
0.02
0.04
0.03
STRAIN
(a)
0.05
0.07
0.06
700
600
-
o EXPERIMENT:TRANSVERSE
PREDICTION: TRANSVERSE
- EXPERIMENT:ROLLING
500
0z
400
-~~~~
300
-
oo065000000
_ --..
0
,0-'
0
w
I-
0
0
200
1001
0
0.01
0.02
0.03
0.04
STRAIN
(b)
0.05
0.06
0.07
Figure H-7: (a) Superelastic experiment conducted along the transverse direction.
The prediction from the finite-element simulations is also shown, and (b) Superelastic
experiment along the rolling direction compared with the transverse direction. The
finite-element simulation along the transverse direction is also shown.
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700
-A- EXPERIMENT:FORWARD
600-
-e- SIMULATION:FORWARD
-*-- EXPERIMENT:REVERSE
-4- SIMULATION:REVERSE
500-
40,
200 --
1- 00 -
0
0
10
20
30
50
40
ANGLE 0
60
70
80
90
Figure H-8: Comparison of the nucleation stresses of the differently-oriented specimens during the forward and reverse transformation at 2% strain with respect to the
prediction from the finite-element simulations.
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Appendix I
Operating procedure for the
RigaKu200 & 300 X-ray
Diffraction machine
Operating procedures for the RU200 & 300:
1. Check the status of the x-ray generator: Determine whether the x-rays are ON
or OFF by observing the tube tower and the RED light on the front of the HIGH
VOLTAGE CONTROLLER. Make sure that the SHUTTER is closed! Do not open
the door in any circumstances if the SHUTTER is not turned off.
2. If the X-RAYS are NOT ON, CHECK the status of the TUBE TOWER VACUUM. The THREE GREEN LIGHTS on the VACUUM CONTROLLER indicate
the quality of the vacuum. If all three green lights are on, then the vacuum is o.k. If
not, report this immediately to the person in charge of the machines.
3. If the X-RAYS are ON determine if either of the X-RAY SHUTTERS are OPEN
by observing the RED lights ON EITHER SIDES OF THE TUBE TOWER. If a
SHUTTER is OPEN, determine whether or not that DIFFRACTOMETER is RUNNING. If neither diffractometer is in the process of COLLECTING DATA, something
is WRONG. X-RAYS SHOULD BE ON and the SHUTTERS OPEN ONLY WHEN
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THE DIFFRACTOMETERS ARE COLLECTING DATA.
4. You may open the diffractometer doors when the X-RAYS are ONLY under these
following conditions: (a) There is a shield in between the two sides, (b) Your shutter
is closed, (c) You must survey your diffractometer area with the hand held Geiger
counter, (d) The intensity of the background around your diffractometer should be
close to that outside the enclosure.
5. If you are the only one using the X-RAY generator, you must turn off the X-RAYS
every time you enter the enclosure to change samples etc.
Start-up procedure:
1. Make sure of the experiment you intend to conduct. Different experiments require different slits to be used. The slits that exist are the DIVERGENCE slits,
SCATTER slits, RECEIVING slits and the SCHULTZ slits.
2. The DIVERGENCE slit is the one closest to the TUBE TOWER. The SCATTER
slit has the same size as the DIVERGENCE slit. The SCATTER slit is the next slit
in the diffracted slit path. It is in the front of the receiving SOLLER slits and is the
next one after the sample.
3. Check the RECEIVING slit. It is located on the back side of the SOLLER slits after the SCATTER slit. Different experiments require different combinations of these
slits the type of slits used will be discussed later.
4. INSERT the SAMPLE. This ia done by mounting the sample on a aluminum
mounter. After mounting it, make sure it is tight against the post. You can use
scotch to anchor the specimen mounter to the post BUT do not anchor the scotch
tape to the STEEL part of the machine because it is stationary.
5. Before starting up the X-RAY, make sure that the SHUTTER is CLOSED! To
start-up the X-RAY generator, press the T-REV button. The AMBER LIGHT on
the HIGH VOLTAGE CONTROLLER should come ON. If the light does not come
on, check that the doors are closed and the KV and Ma selector switches are fully at
their minimum positions. Then press the ON button to activate the X-RAYS.
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6. Turn up the KV slowly and wait for the meter stabilize before turning it up again.
7. Then turn up the MA slowly. The selection of KV and MA depends on you and the
operator of the other machine. DO NOT USE COMBINATIONS WHICH BRING
EITHER METERS TO THE RED ZONE!
8. INITIALIZE the DIFFRACTOMETER. Although this is not necessary, it is done
to assure that the THETA and 2THETA are properly coupled.
9. Now it is ready to take data. Proceed to either the POSITION GONIOMETER
program, the FIXED TIME COUNT or the MEASUREMENT program.
Shutting off procedure:
1. The shutting off procedure is the exact opposite of the starting up procedure.
First turn the MA down.
2. The turn the KV down slowly and turn it down one step a time. Do this after the
meter stabilizes.
3. TURN THE SHUTTER OFF.
4. Press the OFF button.
5. To get the specimen, turn the FAILSAFE key first. You will hear a beeping sound.
Then open the door.
6. Then check the radiation level in the equipment by using the Geiger counter working your way into the chamber. If it is safe enough to do so, take off the specimen.
For 2THETA measurements:
1. Use the 1 degree DIVERGENCE slit, 1 degree SCATTER silt and the 0.3 mm
RECEIVING slit.
2. Repeat the start-up procedure.
3. Once the start-up procedures have been performed, turn the SHUTTER to EXT.
4. Start-up JADE program by C>MENU from the computer.
5. Once the opening window appears go to QUALITATIVE PROGRAMS.
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6. Go to the SET DATUM program. Normally, let the datum remain the same as
before. Exit the window by pressing ENTER until you get out of this window.
7. Then go to the SET HV/PHA program. Normally, the values given on the window
should not be tampered with. Exit the window by pressing ENTER until you get
out.
8. Then proceed to the SCAN CONDITIONS menu. When the cursor appears on
the SET NUMBERS just enter to get to the next menu.
9. Then go to the AXIS menu and make sure that is says 2 THETA/THETA-REFL.
This can be changed using the arrow keys on the keyboard.
10. Then choose the START and STOP angles. The angle ranges from 15 degrees to
120 degrees.
11. Change the SAMPLING INTERVAL to the interval which you want.
12. Leave the DATA TYPE and FULL SCALE range to the default value. To exit
this menu press Ctrl Z. After all of these steps, go to the MEASUREMENT menu to
start data acquisition.
For POLEFIGURE measurements:
1. The divergence slit will remain at 1 degree but the scatter slit must one size
larger than the size of the receiving slit (in mm). The Schultz slit must be fixed as
well. DO NOT FORGET THIS.
2. From the main menu, go to the POLE FIGURE DATA COLLECTION PROGRAM. Do not set the datum but go straight to the SET SCAN CONDITIONS
menu.
3. To obtain the pole figure values, use the beta step to be 5 degrees, beta speed to
be 360 degrees per minute and the alpha step at 5 degrees.
4. Then exit this menu to go to the MEASUREMENT menu to start collecting data
of the pole figure measurements.
5. To obtain the values for the background measurements, the procedure is the same
except that data is taken approximately 2 degrees away from the peaks. This is done
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at a place where the values obtained from the 2THETA measurements are relatively
flat. This is due to the background radiation.
6. Go back to the SET SCAN CONDITIONS in the POLE FIGURE DATA COLLECTION PROGRAM.
7. Change the beta step to 360 degrees with beta speed of 640 degrees per minute.
8. Then go the MEASUREMENT menu to start collecting the data for the background emission.
9. Always remember the name of files you are working on because there are always
many files which are saved into the hard disk.
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